Talk:0.999.../Arguments

From Wikipedia, the free encyclopedia
Jump to navigation Jump to search
Frequently asked questions (FAQ)
Q: You guys talk a lot about real analysis, limits, and calculus; shouldn't this just be about arithmetic?
A: Unfortunately, in order to formally prove many qualities of numbers, one often has to resort to higher mathematics: real analysis in the case of real numbers, number theory in the case of integers, and so forth. The article and arguments page both aim to be understandable to all, but, since many skeptics ask for formal proofs, higher mathematics will inevitably come into play.


Q: Person X made a pretty convincing argument that ! Those who say otherwise are giving arguments I can't understand.
A: Before believing an argument, check sources, responses, and record. The main article is well-sourced, whereas arguments against generally cite (if anything) non-mathematical sources such as online message boards and dictionaries. Also, although most people are trying to write something everyone can understand, some arguments, in relying on higher mathematics, will not be easy to follow for all. Still, try to follow those you can. Finally, those who firmly believe that mainstream mathematics is mistaken will generally reveal their lack of rigor and/or contempt for experts and others who disagree with them. Before replying, read their other contributions to make sure you aren't siding with someone you yourself would not trust.


Q: I have a mathematical question.
A: Please check the FAQ on the talk page.


Q: Do any reliable sources side with the students' stubborn feeling that 0.999... should be less than 1?
A: Yes. See the section on infinitesimals in the main article.


Nominated at MfD[edit]

I have begun a discussion of this talk page at MfD, because it is being used as a forum in violation of Wikipedia policy. This sort of discussion is fun, and there are lots of appropriate places on the Internet to have it. Wikipedia is not one of them. Lagrange613 01:20, 16 October 2014 (UTC)

MFD closed as Keep. — xaosflux Talk 12
58, 24 October 2014 (UTC)

Ellipses denote approximations which ignore infinitesimally small remainders[edit]

.333... is an approximation, valid only to the infinity decimal positions, of the precise value 3/9

Because our calculations will never reach infinity decimal positions we should say that 3/9 ≈ .333..., not 3/9 = .333...

For the sake of postulation, let's suppose that "..." denoted a specific number of decimal positions.

1/9 = .111... with a remainder of .000... followed by the precise fraction 1/9

3/9 = .333... with a remainder of .000... followed by the precise fraction 3/9

9/9 = 1 with no remainder

3 * (.333... with a remainder of .000... followed by 3/9) = 1

3 * (.333... with no remainder) = .999.. with no remainder

3/9 = .333... is a mathematical approximation, it is not an absolute value and cannot be multiplied by 3 to get 1. The approximation times 3 equals the approximation .999..., or approximately 1. It should be written 3/9 ≈ .333...

The use of the particular piece of mathematical short hand that allows 3/9 = .333... is flawed at its core. — Preceding unsigned comment added by 69.51.129.99 (talkcontribs) 18:10, 4 April 2016‎

This is not true within the real numbers, for a simple reason: For every positive real number x there exists a natural number n so that nx≥1 (the Archimedean property). If there were some non-zero difference x between 1/3 and 0.333..., what natural number n would do that? None, since the existence of such a natural number n would imply that there are only finitely many zeros in the decimal representation of x. This leads to a contradiction.
On a more basic level, there seems to be a misunderstanding regarding the meaning of "..." and infinite decimal representations. There are no "infinity decimal positions" that are treated differently in principle from the first, second, third and so on decimal positions. The "..." in 0.333... means that in each of the infinitely many decimal positions, each of which is itself finite (since there are infinitely many finite natural numbers), is a "3". Huon (talk) 19:20, 4 April 2016 (UTC)
I'm not entirely sure I follow the IP's post, but it seems to me that what it may have intended is that the expression 0.333... means that you write down some large-but-finite number of 3's. He/she may reject the idea that having infinitely many 3's after the decimal point is even possible. That is a possible position to take (see ultrafinitism). I'm not sure what, if anything, the article should say about the ultrafinitist view (presumably ultrafinitists could still allow an expression like 0.3̅that's supposed to be a 3 with an overbar; didn't come out as well as I hoped on my screen, and reason about it formally). --Trovatore (talk) 20:04, 4 April 2016 (UTC)
(And this was so beautifully quiet for so long...)
I confess that I really don't understand what the IP is trying to get at. If Trovatore has read him/her correctly, s/he thinks that the expression 0.333... must have only a finite number of threes, because if you start writing them out, one at a time, you'll never reach a point where there are an infinite number of threes. But there is no need to have an infinitieth 3 (which is good because there isn't one). You just need it to be such that if you consider any specific 3, there is always one after it. Like I said above: imagine you are immortal. You never actually reach an infinite age (since your age is a real number), nor do you need to. You just need to be 100% certain on every day that you will live to see the next one. Double sharp (talk) 17:06, 16 April 2016 (UTC)
Well, it depends on how you're understanding the argument. To me, the most straightforward approach is to take the eight characters "0.999..." to be a shorthand for a string containing infinitely many 9s. Then you argue that the interpretation of that string as a real number gives you precisely the real number 1.
The OP seems to be saying that no such infinitely long string exists, so 0.999... is shorthand for something that doesn't exist, and therefore is not a meaningful expression at all.
You on the other hand seem to be saying that you don't have to accept a completed infinite string of 9s, because the limit can be shown to be 1 without them. That is true, but now the story gets more complicated — 0.999... is no longer shorthand for an infinite string, but instead has to be taken to be eight literal characters which now have to be given an interpretation. It would be something like a specification for a computer program that, given a number n, returns the first n 9s. Then you have to give an account of the interpretations of such computer programs, which makes you address details that seem to be a bit of a distraction. (Also, it would be limited to the computable reals, and most reals are not computable.)
Note that a completed infinite string of 9s is not at all the same thing as an infinitieth nine. --Trovatore (talk) 20:19, 17 April 2016 (UTC)
I've always understood infinities in mathematics as processes rather than objects, which matches your concept of a computer program specification. The 0.999... is an infinite process because, no matter how many 9s you have added, you can always can add one more; this is not true of many other mathematical process which are finite and restrict how many times you can execute them (such as subtracting 1 from a natural number to get another natural). Thus, "infinite 9s" means "in any finite string, you can always have at least one more"; and that infinite process certainly exist.
Now, to make that string equal to the number 1, you have to define the limit of the process as the "smallest number above any of the partial, finite strings of 9s". Only when you make such definition you have the equivalence of the process with a number, any number, which in the reals happens to be the number 1. Diego (talk) 11:12, 18 April 2016 (UTC)
@Trovatore: I don't personally have a problem accepting a completed infinity myself, but given that the OP seems to be rejecting the existence of a completed infinity of 9s, I felt that it would perhaps be better not to insist on it in my argument. Double sharp (talk) 12:12, 18 April 2016 (UTC)

It seems that one of the biggest hangups behind most people who don't accept the equality 0.999... = 1 is that they expect that you can start from the beginning and number the nines starting 1, 2, 3, getting every positive integer along the way (this is fine so far), and then (this is the crucial misunderstanding), getting to a last nine numbered ∞ (though I guess it ought to be called the ωth 9 instead). Such a thing does not exist in the real numbers. And even if it did (which seems to move straight into the hyperreals), we run into another problem: if this ωth 9 comes after all the 9's numbered with positive integers, what comes after the ωth 9? Shouldn't there be an (ω+1)th 9 as well? And an (ω+2)th? And so on. So even if you would allow hyperreal-style infinite nines, the hyperreal with nines going all the way (analogous to the one real with nines going all the way) is still one (because there is no first thing greater than zero, even in the hyperreals). (Assuming my understanding of hyperreals is correct, since I only started on that recently.) Double sharp (talk) 05:56, 17 April 2016 (UTC)

Why is there so much talking about it?[edit]

Consider the difference 1-0.(9). If you do this subtraction left to right, this difference is ten times smaller on every new step than it is on the previous step: 1, 0.1, 0.01, and so on. The difference is no more than any number that you achieve in this sequence, and therefore less than any previous number in this sequence. The difference is less than any positive number and more than any negative number. But this is exactly what zero is. Now, if the difference of two numbers is zero, then how on Earth these numbers can be anything else than equal? That's what I don't understand. - 91.122.7.245 (talk) 14:05, 27 April 2016 (UTC)

Sure, any number that has zero, many nines after the dot, and then only zeros is less than one. But this is not the number that is denoted by the notation, because it has all zeros and not all nines at the end. If we can't reach the limit, then we have a totally different string of digits, and the question is different. So, I just don't understand what causes the confusion this time. Lack of imagination? - 91.122.7.245 (talk) 16:17, 27 April 2016 (UTC)
In my experience a common misconception is that people think of 0.999... as some kind of process that never quite "reaches" 1. Others argue that the difference should be some non-zero infinitesimal and are more willing to abandon the real numbers than accept that 0.999...=1. Huon (talk) 00:26, 28 April 2016 (UTC)
So, the confusion seems to be about the denotation… What does the denotation mean, what is that object whose properties are to be investigated… The “process” is a rather unclear object, because it's not static and is not all in vision, but the root of the confusion is clear, it seems… I. e., like with the diagonal argument, the cause is the wrong question again: a question that is asked about an incomplete state of the things which cannot be static and therefore cannot provide an answer. (Like just one real number to enumerate instead of the complete set that needs to be enumerated.) The reason why I was wondering was that I didn't believe that some people had better logical abilities than others. If someone insists to be wrong, it's probably not a failure of the “grasp of elementary notions“ when developing an answer, like Trovatore suggested somewhere, but a failure to ask oneself the right question… - 91.122.0.103 (talk) 14:49, 28 April 2016 (UTC)
That confusion is inherent to everything labelled as "infinite" in math. Does an infinite set exist if it can't be computed? Is there "a member at the infinite"? Does it have an infinite amount of members, or is it just that you can compute any finite member? These questions are intuitive and reasonable to ask, but in the end the only effective way to handle them is to use a formal approach and ask "what axioms define the properties of the the infinite object?" Depending on the axioms chosen, the answers to those questions may vary. Diego (talk) 15:25, 28 April 2016 (UTC)
While different questions, if they are correctly made and concern static objects, can indeed yield different static answers, I don't believe that these questions must necessarily be formulated in a formal language to yield meaningful answers. Perhaps a formal language is just a convenient tool of communication for mathematicians. But this is, of course, a very different question. And I am not prepared to go in the depth of details, because I am not a mathematician… - 91.122.0.103 (talk) 15:36, 28 April 2016 (UTC)
You can formulate them with the language of philosophy as well, which resembles natural language. But in the end, to resolve ambiguities you need to reach a level of detail not different from formalism. Using natural language merely gets you some shortcuts at the steps where precision is not required, but it can be tricky to assess where those shortcuts can be made safely. Diego (talk) 15:57, 28 April 2016 (UTC)
P.S. I like your description of the differences getting smaller and smaller until they "disappear" below any positive number you may think; it reverses the common misunderstanding of "always having a small but non-zero amount". In fact, that is how the limit of the sequence is defined formally. Diego (talk) 16:01, 28 April 2016 (UTC)
While the level of detail may need to be the same with the two approaches to exposition, the method of exposition of that detail is different. I don't think that resolution of ambiguities necessarily makes formal the language to use. Basically, the question is: does my thinking happen naturally and without myself being aware how it really happens, or I need to put something in paper in correspondence to my thought as it should happen? In the first case, the use of formal language is not a pre-requisite to get something right: I just use natural language to only point at the thought process rather than mirror it in part or in whole, like it happens in real life, too. Rather, the pre-requisite is to ask the right question, as one's mind does its own independent work to arrive at questions and find answers. However, the use of natural language may probably be tedious for large volumes of mathematics… - 91.122.0.103 (talk) 18:15, 28 April 2016 (UTC)

Interpretation within the ultrafinitistic framework[edit]

See here:

" Why the "fact" that 0.99999999...(ad infinitum)=1 is NOT EVEN WRONG

The statement of the title, is, in fact, meaningless, because it tacitly assumes that we can add-up "infinitely" many numbers, and good old Zenon already told us that this is absurd.

The true statement is that the sequence, a(n), defined by the recurrence

a(n)=a(n-1)+9/10^n a(0)=0 ,

has the finitistic property that there exists an algorithm that inputs a (symbolic!) positive rational number ε and outputs a (symbolic!) positive integer N=N(ε) such that

|a(n)-1|<ε for (symbolic!) n>N .

Note that nowhere did I use the quantifier "for every", that is completely meaningless if it is applied to an "infinite" set. There are no infinite sets! Everything can be reduced to manipulations with a (finite!) set of symbols."

Count Iblis (talk) 22:32, 13 July 2016 (UTC)

And he is doing very poor mathematics as he rejects the axiom at hand to refute the statement within those axioms, hence his statement is not even wrong. It is just stupid. Just claiming "infinite sets don't exist" is wrong, because they do in mathematics wether he likes it or not, they are defined to be as anything else. TheZelos (talk) 14:45, 14 February 2017 (UTC)

Blindly accepting decimal as the representative numeric system for all numbers and situations[edit]

Just a quick thought. Having .999~ represent 1 bases itself on the assumption that .333~ is the correct representation of 1/3. (We can not and do not have an accurate representation of something between a 3 and a 4 when writing decimal numbers)

This kind of fractions are what I would consider a defect of the decimal number. So we just have an article about one of the artifacts of a deficient numeric system(deficient at least for the task of dividing and multiplying by 3).

On ternary system .333~ or 1/3 would be .1, on a 30 digit system it would be 0.a and we wouldn't be having dumb articles like this. — Preceding unsigned comment added by RenatoFontes (talkcontribs) 21:25, 20 July 2016 (UTC)

I have moved this here from the talk page. KSFTC 21:38, 20 July 2016 (UTC)

Try to figure out the base-30 representation of 1/29. Now multiply by 29. --Trovatore (talk) 22:22, 20 July 2016 (UTC)
True: decimals are not numbers. The fact that decimals represent numbers is actually a theorem. But it is not written down in stone that every number corresponds to one and only one decimal. That would actually be wrong: the subject of the article is an example of how and why this is wrong. Sławomir Biały (talk) 22:34, 20 July 2016 (UTC)
The same concept applies to any number base. For instance, in base-16, 0.FFF... is equal to 1.—Chowbok 05:02, 18 September 2017 (UTC)

.333... x 3 = 1, NOT .999... What does this imply?[edit]

1 ÷ 3 = .333...

therefore

.333... × 3 =1 (NOT .999...)

So wouldn't that imply 0.999... and 1 are different things? Not saying that this proves 0.999... < 1, but just that t's just something else.

Are there calculations that give a result of 0.999...? That is, one where we are compelled to give the answer explicitly as "0.999..."? If not, it seems as though "0.999..." only exists fictionally for the sake of us arguing about it. --96.35.2.199 (talk) 18:32, 12 September 2016 (UTC)

You could say that as well of any infinite number, like "pi" or "e"; there's no way you can write them in full. That doesn't make them any more or less "existing" nor "fictional". Infinite numbers are typically described by the operations used when manipulating them through arithmetic and calculus, not by enumerating them to the end. In this case, 0.333... x 3 is clearly = 0.999... as can be seen from the basic digit-by-digit calculation, and it's also clear that 0.333 x 3 = 1 as well (since it's the inverse of 1 ÷ 3 ). Diego (talk) 21:51, 12 September 2016 (UTC)
Thanks but I think you misunderstood part of my question. I'm not talking about writing the entire expression out in full with an endless string of 9s. What I meant was, when would one need to give the representation "0.999..." as a result? In other words, why in any practical application would one feel the need to write out a zero, a dot, 3 nines and 3 dots when the result can just be given as "1"? --96.35.2.199 (talk) 23:40, 12 September 2016 (UTC)
By definition 0.999... is the sum . This is an infinite sum that has a meaning independently of whether it is equal to one or not. It happens to be a mathematical theorem that this sum is equal to one. But is meaningful apart from its 1ness. Sławomir Biały (talk) 00:21, 13 September 2016 (UTC)

Article is overtly biased toward the veracity of 0.999... = 1[edit]

This article is propagandist and does not adopt an unbiased view of the subject "0.999...". 1. It offensively belittles "students" as the group mostly holding to the "wrong" view that "0.999... < 1", with complete disregard to the possibility that the intuitive result may be right.

2. It overtly treats proofs supporting the "right" result more favorably than proofs supporting the "skeptical" result. The very word "skeptical" is used in the overall tone of the page as a pejorative. The correct headings would be "Arguments supporting "0.999...=1" and "Arguments supporting "0.999...<1" with equal treatment of both.

3. I have attempted to add a reference to a blog post containing a robust (and I might add, formidable) proof that very clearly demonstrates (possibly rigourously) in elementary school math that "0.999... < 1". This reference has been excluded on the basis that (in the excluder's opinion) the poster "does not understand limits". That may well be the case, however, the proof makes no reference to limits and has not dependency upon them. My reading of the blog post is that the discussion regarding Limits" is merely an opinion piece to promote debate, not offered as any part of the proof. Consequently, the reason for exclusion is both spurious and irrelevant, and simply reinforces my feeling that this page is far from objective. Alex Alexander Bunyip (talk) 15:07, 24 July 2016 (UTC)

It's a theorem that the real number represented by the infinite decimal expansion 0.999... is identical with the real number 1. There are high quality sources that have proofs of this, beginning with the axioms of the real number system. As a mathematical theorem, a disproof would essentially imply that all of mathwmatics involving real numbers is inconsistent. One of the pillars of Wikipedia is WP:NPOV, which in particular implies that subjects like this are discussed according to the weight of different viewpoints in reliable sources. There are various sectioning of the article that discuss septicism, alternative number systems in which 0.999... is different from 1. Sławomir Biały (talk) 15:21, 24 July 2016 (UTC)
@Abunyip: We rely on reliable sources as our primary basis for weighing claims in articles. The claimed proof you cite is on a self-publishing website, and thus does not count as being a reliable source. This is to be contrasted with the many proofs given in reliable sources which demonstrate that 0.999... = 1, which provide the basis for the article's presentation of 0.999... = 1 as established mathematical fact. Please do not re-add the material without providing a reliable source that supports it. (Also: have you see this proof, which demonstrates 0.999... = 1 from first principles?) -- The Anome (talk) 15:41, 24 July 2016 (UTC)
I think we should include a sentence about the formal proof, citing the metamath source. Also, a standard challenge to anyone claiming to have discovered a "watertight" proof that 0.999... ≠ 1 should be "ok, well formalize your proof in metamath" (or Coq or HoLight, etc) Sławomir Biały (talk) 16:09, 24 July 2016 (UTC)
I'm not sure we can cite Metamath directly, as it's not a WP:RS of itself, which is why it's in the external links section rather than the body of the article itself. Are there papers on Metamath that we cite in its stead? -- The Anome (talk) 16:22, 24 July 2016 (UTC)
But yes, inviting people to formalize their argument would go a long way to helping clarify things. Not least for them themselves. -- The Anome (talk) 16:37, 24 July 2016 (UTC)
I would consider Metamath to be a reliable source. It does not seem like a theorem proven in Metamath is likely to be challenged. Indeed, Metamath is probably much more reliable than many textbooks, etc. Sławomir Biały (talk) 17:47, 24 July 2016 (UTC)
The article should be biased in favor of the viewpoint that 0.999.. = 1, because that is the viewpoint of essentially every mathematics reference. The idea of neutral point of view does not mean that we are neutral between all viewpoints; it means that we are neutral between viewpoints to the extent that they are represented in high-quality sources. The viewpoint that 0.999... is the same real number as 1 is so overwhelmingly dominant in the mathematics literature that, even if some other viewpoint might be possible, this article should reflect the viewpoint that the numbers are equal. A better question to ask might be: why do so many sources say that 0.999... is equal to 1? What do they mean by "equal"? That will help clarify what is going on in the literature. — Carl (CBM · talk) 16:11, 24 July 2016 (UTC)
There is no such thing as a "disproof" of this equality. Once you have a correct proof one way, there cannot be a proof contradicting that proof using the same assumptions. @Abunyip: I advise you to read more on what a proof is. The "source" you cited is not only unreliable, but the poster clearly does not know what they're doing, because he blatantly fails to use a correct definition of convergence of a real sequence. Either that, or he's taking a fringe, unaccepted way of looking at real analysis.--Jasper Deng (talk) 16:55, 6 October 2016 (UTC)

There is a correct proof that every 9 fails to reach 1. It is so by definition. What else do you want? If there is a counter proof, then the theory is useless. — Preceding unsigned comment added by 84.155.143.190 (talk) 17:30, 6 October 2016 (UTC)

@84.155.143.190: But that proof is wrong, because that's not the meaning of convergence. One of the fundamental properties of the reals is that between any two reals, there's another. There is, however, no real number between ".999999999..." and 1.--Jasper Deng (talk) 17:32, 6 October 2016 (UTC)

Of course the sequence 0.999... converges to 1, but being a sequence, it is not equal to 1. Having limit 1 and being equal to 1 are two different things. — Preceding unsigned comment added by 84.155.143.190 (talk) 17:38, 6 October 2016 (UTC)

But .999... has to be understood as the limit of the corresponding sequence. It has no meaning as a real number.--Jasper Deng (talk) 18:01, 6 October 2016 (UTC)
Why not tell the truth? 0.999... is a sequence. It has no numerical value, but it has a limit. Writing 0.999... = 1 is sloppy, confusing, and lacking the precision required in mathematics. Further, according to set theory there are all terms. You cannot denote them if you use the correct notation for the wrong notion, i.e., the limit. These things should at least be described in an unbiased article. — Preceding unsigned comment added by 84.155.143.190 (talk) 18:16, 6 October 2016 (UTC)
And finally every mathematician can verify that 0.9, 0.99, 0.999, ... is abbreviated by ...(((0,9)9)9)... and this is abbreviated by 0.999... The first one is not understood as its limit. Why should the last one be? — Preceding unsigned comment added by 84.155.143.190 (talk) 18:19, 6 October 2016 (UTC)
No, we write repeating decimals to represent rational numbers that happen to be the limits. That's the way positional notation works, and the way it is understood. There's nothing ambiguous about that. Other notations of the same sequence might not be understood as such, but that's the way this notation is interpreted, period.
This article is fine as-is. The very first sentence of the article says that's the way we read something like .9999999999 or .33333333333.--Jasper Deng (talk) 18:41, 6 October 2016 (UTC)
I agree strongly with the revert of this edit. The article contains quite a few high quality references supporting the contention that 0.999... = 1. Without countermanding sources of equivalently high quality contesting this identity, it would be inappropriate to call it "erroneous" in Wikipedia's voice. If there are reliable sources that contest the identity of 0.999... and 1, then we can reference those in the article, being careful to emphasize their WP:WEIGHT appropriately. For the record, I actually think that the current article does a good job of accommodating dissenting viewpoints. Even when such views might fall on the wrong side of WP:FRINGE, they provide an interesting and balanced article. But we need references of a sufficiently high quality to merit inclusion, and very high quality references indeed are required to put anything into the lead (such as a standard textbook on Real Analysis, for example). Sławomir Biały (talk) 20:03, 6 October 2016 (UTC)
I agree.--Kmhkmh (talk) 00:02, 7 October 2016 (UTC)
Is a text book published by one of the biggest science publishers "high quality" enough? — Preceding unsigned comment added by 84.155.136.151 (talk) 06:55, 7 October 2016 (UTC)
It probably would be. But it would actually have to discuss the subject of this article. As far as I can tell, the book you cited earlier did not. Certainly, neither of your main points that Euler "erroneously claimed it" and that it "looks true to someone with a sloppy mind" seems likely to appear in a reliable mathematical source, and do not in the source you cited earlier in this discussion page. Finally, any source would need to be weighed against the other high-quality sources to see if the views it contains are appropriate for the lead of the article (which is where the edit under discussion is). The current article has many high-quality mathematical sources containing proofs that 0.999... = 1. Only if the dissenting sources carry a comparable weight to those in the current article can a view be added to the lead, per WP:FRINGE. Sławomir Biały (talk) 10:44, 7 October 2016 (UTC)
0.999... is not a limit and not a sequence. The pedagogical section of the article does not seem to be prominent enough. Hawkeye7 (talk) 21:04, 7 October 2016 (UTC)
This comment is puzzling. I agree that the literal string of symbols "0.999..." is not a limit. It is a zero, followed by a period, followed by three nines and an ellipsis. But the real number represented by this notation is a limit, namely the value of the infinite series . Without clarification, I have no idea if this is what you mean, though. To pose a question: if it's not a limit, then what is it? Sławomir Biały (talk) 22:10, 7 October 2016 (UTC)
Moreover notation is also a question of convention and the literature i've seen treats as notation for (the limit of) that infinite sum.--Kmhkmh (talk) 02:56, 8 October 2016 (UTC)
It's a real number. It's called "one". As you say, it is a matter of convention. We can write it as ١ or 1 or . It is the value of the infinite sum . Hawkeye7 (talk) 03:52, 8 October 2016 (UTC)

It's more than a matter of convention. The number is not by definition equal to one. It is a mathematical theorem that it is equal to one. The definition of this number is as a limit: the sum of an infinite series is one type of limit. It can be proved that the value of this limit is identical to the real number 1. and so the two numbers are equal. But it's really misleading to say that is "not a limit". It is a limit. It is also one. Sławomir Biały (talk) 12:05, 8 October 2016 (UTC)

Yes, as the article says:
The first two equalities can be interpreted as symbol shorthand definitions. The remaining equalities can be proven.
So simple. - DVdm (talk) 20:46, 8 October 2016 (UTC)

————— — Preceding unsigned comment added by 2601:188:4101:D000:C184:3F0B:716A:781B (talk) 04:56, 29 December 2019 (UTC)

I have to agree with the original criticism. This article is ideological propaganda (which is common here) in favor of mathematical platonism that intentionally or not misrepresents the problem. This question of whether .999... = 1 is the canon example, and litmus test, of the conflict over the foundations of mathematics between the schools (a) demanding the scientific basis of mathematics (mathematical realism) by Hilbert and (b) the literary (pseudoscientific) basis of mathematics that was reintroduced by Cantor resulting in the catastrophe of mathematics, logic, and even mathematical physics in the twentieth century. So it is not a question of pedagogy but an unsettled conflict over the choice between mathematical realism under which no infinity is operationally impossible, limits always extant in any application, and therefore .999 != 1, versus mathematical platonism dependent upon the law of the excluded middle, under which deductively, one cannot construct a statement in the vocabulary and grammar of mathematics (the logic of positional names) where .999... does not equal 1. This is the battle between realism (science, operational mathematics), and idealism (philosophy, literary mathematics).

For example, Descartes was important because he restored mathematics to geometry (operations) giving us the cartesian model, and the result was newton-liebnitz's calculus on one end and the restoration of the realism on the other. Cantor, Bohr, and yes, even Einstein as well as the logicians tried to restore idealism. This led to the constructivist argument. That argument succeeded in physics and has slowly propagated through the sciences, even, oddly causing the reformation of psychology (although not sociology). Computer science has taken up constructivist mathematics leaving mathematical platonism to the discipline of math. Unfortunately, we are stuck with Einstein-Bohr-Cantor versus Hilbert-Poincare-Turing, and this is one of the profound failings ofthe 20th century.

For example. Numbers exist as names of positions and nothing else. We use positional naming to generate unique names. Positions are ordered but scale independent. All of mathematics consist of functions producing names in the grammar and vocabulary of positional names. Cantor states that we can produce multiple infinities of different sizes. This is a fictionalism (parable). Instead, no infinity is constructible only predictable in imagination. So, in any sequence of operations, different sets will produce new positional names at different rates, such that at any given limit, the sets will differ in sizes. There are no different 'sizes' of infinities, only different rates of production of positional (unique) names. Math is full of such parables.

In ethics for example, the litmus test is blackmail: it's voluntary, it's an exchange, but why do we react against it? Because it's an unproductive transfer. In logic it's whether logic is binary and a rule of inference (true vs false) or ternary and scientific (false, truth candidate, undecidable). In mathematics the litmus test is whether .999... = 1. Under realism, no it doesn't. Under idealism (Platonism) it does. Science (meaning testimony) imposes a higher standard than idealism (platonism). Platonism remains justificationary and Realism falsificationary.

So when you make the claim the question is pedagogical (error) and that people don't understand - that's patently false. It's that operationalism (realism, science) has a higher standard than platonism (idealism, prose). And under realism .999... cannot possible ever equal 1 since no infinity is operationally possible. Whereas under idealism the standard is lower, because under scale independence, infinity substitutes for the unknown limit, which as a consequence is 1.

The fact that people aren't pedagogically informed that this debate exists, and persists, and that its origin is between western engineering and geometry, and middle eastern algebra and astrology, leading to western reason and science, versus eastern theology and mysticism - then you begin to understand how important this question is - and why our physicists have been lost in mathematical platonism - and why scientific woo woo is so common, when it's increasingly likely that mathematics of positions names (points) has most likely reached its limits. And that we have failed to create the next generation of mathematics (shapes, geometries) that would allow us to solve protein foldings and the structure of the universe that results in our observed but unsolvable quantum distributions of probability.

Cheers

2601:188:4101:D000:C184:3F0B:716A:781B (talk) 04:54, 29 December 2019 (UTC)

As the comment immediately above yours tells you, the string "0.999..." is shorthand for a mathematical limit that can be proven to have the value 1. It effectively —and only— means that "'the more nines you write, the closer you get to one." I'm sure that everyone agrees with that. Everything else is balast. - DVdm (talk) 10:22, 29 December 2019 (UTC)
It's not 'proven' scientifically (surviving falsification under discovered laws), just the opposite - it's falsifiable and falsified scientifically (the universe provides the only closure). It's only internally consistent (demonstrated by proof using declared axioms) using a pseudoscientific presumption: the excluded middle, where the excluded middle demarcates the conflict between realists (Scientists) and platonists (fictionalists), by a fictionalism of closure, when the 20th 'proved' there is no closure in any axiomatic system. The 'number line' is a fictionalism it doesn't exist. An infinite series is impossible. Infinity doesn't exist. So no. As I said, this is the canon example of the conflict between mathematical platonism (pseudoscience) and mathematical realism (science). Existentially, a number no matter its expression is the name of a position as a ratio of an identity ('one') produced by a series of functions. And therefore the article is ideological not NPOV. Under NPOV, the answer to the question of whether .999... = 1, is dependent upon mathematical platonism (fictionalism) or mathematical realism (science). As far as I know, science is the standard for truthful speech, and the NPOV. Theologians maintained, like platonists maintain, justificationary nonsense, rather than reform. Math needs a reformation because like many topics, the late 19th and early 20th restored fictionalisms despite the efforts of the empiricists and the scientists, and they were able to do that through sophistry in mathematics, made possible by the tradition of platonism. And your comment is evidence of the problem. "We can get away with it." Same way Niels Bohr could equate the idealism of quantum mechanics without solving the underlying operationalism. This is, one of the most important problems of the age, and the conflict between scientific and fictional mathematics like the debate between aristotelian and platonic philosophy remains the principle impediment to the unification of the fields under a single paradigm consistent, coherent, and complete.
Cheers
2601:188:4101:D000:C184:3F0B:716A:781B (talk) 15:34, 29 December 2019 (UTC)
Sorry, but it is proven, mathematically. And perhaps you are not aware of it, but mathematics is not a science, by definition :-) - DVdm (talk) 15:53, 29 December 2019 (UTC)
Which equates to 'but it is proven theologically', which is a special pleading (look it up) meaning it isn't proven, it's not true, and you've just illustrated my point. ;) So (a) special pleading, (b) false equivalency (intentional ambiguity), (c) private langauge. The debate is between Mathematical platonists and Mathematical Realists (Scientists). And requires disambiguation not false assertion that violates NPOV. In other words mathematical platonists have no claim to ownership, decidability, or truth of the logic of positional names, only to the habits (conventions) and 'private language' of a discipline. One has to additionally INVENT falsehoods (fictions) in order to make the claims. Now you are welcome to find a world authority on the subject to disagree with me (I probably know them) and they will say this "Truth is a matter for philosophers and science, in mathematics we deal only with proofs, where a proof consists of satisfaction of deductibility under the presumption of the law of the excluded middle.". I don't err. Sorry. You're just chanting sophistry by special pleading. Ergo, if you practice mathematical platonism (fictionalism) then you can claim internal consistency. But you cannot claim you speak the truth. So again, the disambiguation is this: that under mathematical platonism (mathematical fictionalism) - you can look that up - .999... is presumed to be equal to 1, wherein, under mathematical realism, .999... cannot be equal to 1. That's the correct disambiguation. Idealism = scale independence, and Realism != scale independence. Find an authority that disagrees. (You won't).
Sorry. Just how it is. Deal with it. NPOV requires disambiguation, not pretense (ideology).
2601:188:4101:D000:35C4:6C94:4DB6:C174 (talk) 21:56, 29 December 2019 (UTC)
I think this is how it is: the article 0.999... starts with "In mathematics, 0.999... denotes..." Deal with that. And, if indeed you don't agree that the more nines you add, the closer you get to one without ever reaching it, then that's... well, let's say, just unfortunate. Try to endure the bafflement Face-wink.svg. - DVdm (talk) 10:27, 30 December 2019 (UTC)
Stop lying by denying please. Sophistry is tedious. The intellectually honest, fully explanatory, coherent, correspondent, complete, and therefore correct (Truthful) definition of the argument is "In Mathematical Platonism .... whereas in in Mathematical Realism .... ". As such the article requires disambiguation. Otherwise you are making Theological or Philosophical rationalization (excuse) for persisting a falsehood by denial. There is a vast literature on the various attempts at a foundation of mathematics. The logicians have settled on a set-theoretic (ZFC) and the realists on an operational. You are welcome to find some authority that disagrees with me but you won't find one. (I know so because I'm one of the authorities on the demarcation question.) An entry level discussion is here: https://en.wikipedia.org/wiki/Foundations_of_mathematics#Foundational_crisis and this page lists most of the spectrum of choices. This page currently asserts a truth that is only a bias, and your argument asserts that 'mathematics' consists in your interpretation, by this bias, when, the evidence states quite clearly, that the conflict on the foundations of mathematics and therefore the answer to this question, which is the litmus test of the differences between those foundations, remains open despite the failure of the logicians in the 20th century and the end of the analytic program. — Preceding unsigned comment added by 73.114.18.178 (talk) 16:26, 30 December 2019 (UTC)
Please sign all your talk page messages with four tildes (~~~~) and indent the messages as outlined in wp:THREAD and wp:INDENT. Assuming that you replied to my message, I have increased its indentation level. — See Help:Using talk pages. Thanks.
So I take it you don't agree that the more nines you add, the closer you get to one without ever reaching it. Okay, no problem Face-smile.svg. - DVdm (talk) 18:31, 30 December 2019 (UTC)
Stop wasting my time with sophistry. Either (a) the debate over the foundations of mathematics exists and (b) the answer to the question is determined by whether one arbitrarily chooses the ideal, platonic, and supernatural, or the real, Scientific, and operation, or it doesn't. Evidence is I am correct. If you had an argument you would cite sources. You don't. You can't. The page must be disambiguated. Sorry. I don't have time for juveniles. 73.114.18.178 (talk) 18:51, 30 December 2019 (UTC)
The sources for the limit are given in the article. I think that the one who is wasting their time is you. - DVdm (talk) 19:14, 30 December 2019 (UTC)
@73.114.18.178: I am not going to argue the merits with you, but I think you may profit from knowing that some of the terms you are using are usually understood in a different way than you appear to be using them.
Mathematical realism, as the term is standardly used, is not opposed to mathematical Platonism; rather, the latter is a particular form of the former. Realists, in a philosophy-of-math context, hold that mathematical objects are real (hence the name). In most cases they do not hold that mathematical objects are physical, and therefore they do not subscribe to physicalism or materialism. Per our articles, in addition to holding that mathematical objects are real, Platonists also hold "that mathematical entities are abstract, have no spatiotemporal or causal properties, and are eternal and unchanging"; these considerations make Platonism more specific than realism in general, but nevertheless a form of it.
Platonists are also not fictionalists; fictionalists (in a phil-of-math context) hold that mathematical objects are "useful fictions", whereas Platonists hold that they are real. --Trovatore (talk) 19:29, 30 December 2019 (UTC)
A sophomoric distinction without a difference. Whether Idealism, Platonism, or Fictionalism on the one side or Realism, Intuitionism, or Operationalism on the other, the argument remains the same. This page overstates the case (is false) because the foundations of mathematics remain in dispute, and the litmus test of whether .999... = 1 or not is determined by whether one relies on imaginary and verbal (philosophical justification) in the platonic and tradition, or demonstrated and actionable (science and falsification) in the Aristotelian tradition. The correct representation of the question is to edit the page to inform the audience of the reason for the dispute (it's not ignorance, it's not an error in judgement) it's a fundamental dispute over how a textual statement in the grammar and vocabulary of positional names (which isn't disputable) is interpreted either under the competing factions of mathematical theorists. A similar litmus test is the Liars Paradox, wherein a fictionalist or hermeneuticist finds a paradox, and a scientist(Realism, naturalism, operationalism) finds only bad grammar failing the purpose of grammar: continuous recursive disambiguation. Another is social construction. The fictionalist (you) argues that social construction is equal to Truthful (consistent, correspondent, coherent, and complete). So my question is, why do you want to preserve a lie rather than just disambiguate the question truthfully? I mean, shallow wits and malinvestment in error in pursuit of self image is what it is. But the problem is rather obvious, just as the liar's paradox and social construction are very obvious: they're lies. So. Why not tell the truth? Why lie? 73.114.16.126 (talk) 02:56, 31 December 2019 (UTC)
@73.114.18.178: You are using the word "realism" wrong, which is the main point I was getting at. Platonists are not opposed to realists. Platonists are realists. I think the word you probably want is "materialism" or "physicalism" (or possibly "nominalism") rather than "realism", though this is a bit speculative as you have not made your position clear enough to be certain. --03:30, 31 December 2019 (UTC)

@73.114.16.126: If you had any reliable sources that support this interpretation, then present them here. Otherwise, as has been repeatedly stated here, per WP:DUE we won't cover it at all. Right now, all I see is hand-waving, specifically using big words without actually saying anything substantial that we can add to the article. The view presented is that prevailing in reliable sources. If there were really two "factions", then the article would look more like zero to the power of zero. Absent a plethora of independent reliable sources that support this view, we will not discuss it here.--Jasper Deng (talk) 03:07, 31 December 2019 (UTC)

Reliable sources? You mean like Goedel? And what is this about 'big words'? You can't follow the (obvious) reasoning and your argument is an ad hom to obscure your lack of comprehension? Fine. I'll collect overwhelming number of sources. But why is it, that I'm absolutely certain, you'll double down on priors to preserve your malinvestment in a falsehood? It's impossible to get a PhD in the field and not know the disputes continue, that there are numerous factions, and that these factions fall into no less than pure (ideal) mathematics and applied (real) mathematics - or that ZFC vs say, Type Theory remains open because of the Axiom of Choice and the fiction of infinity, and the questionability of sets. So what you see is my disbelief that you would offer an opinion on a subject while demonstrably lacking any knowledge of the subject. Answering the question for readers is quite simple: the choice of real or ideal is arbitrary. Yet that choice determines the decidability of .999... = 1 or not. If cites are required, cites we shall produce. 2601:188:4101:D000:3D2B:EC6C:1558:EF68 (talk) 15:13, 31 December 2019 (UTC)
Yes, please produce cites that—for once and for all—invalidate all the known proofs that the more nines you add, the closer you get to one without ever reaching it. - DVdm (talk) 16:19, 31 December 2019 (UTC)

Infinitesimals[edit]

Is an infinitesimal the same as an infinitely small number? Is there a dispute whether it counts as a "number"? I had come up with the idea 1-1/∞=.999... on my own and was suprised to see it here in a slightly different format in 0.999...#Infinitesimals.--User:Dwarf Kirlston - talk 02:31, 15 February 2017 (UTC)

The defenders here confuse the "Real" in real numbers into meaning that that is the "right" number set and the others are all "wrong" somehow. This is why they constantly introduce real-set assumptions into discussions with people who are plainly not talking about Reals in order to bog down the discussions and discourage them. Algr (talk) 20:14, 10 June 2017 (UTC)
The defenders here are fully aware that we are discussing the equality of 0.999... and 1 within the language of discourse of the real number system, in which in infinitesimals simply do not and cannot exist. There are an infinite number of number systems that can be created in which the equality does not hold (my own assertion right here, bet it's true.) The first sentence of the article makes it entirely clear that we are working within the real number system. --jpgordon𝄢𝄆 𝄐𝄇 14:59, 19 July 2017 (UTC)
That really depends on what you mean by 0.999... . If you mean a number which has infinitely many nines after the decimal point, then in most such systems (e.g. in hyperreals - i.e. a non-standard model of the first-order theory of real numbers with infinite and infinitesimal values) there is no such unique number, but an infinite amount of numbers each differing from 1 by an infinitesimal. But if you define 0.999... as the single number that has a nine at every decimal point, then even then 0.999... is equal to 1. (We could instead try defining mathematical operations on the decimal expansions themselves, but this isn't very useful: if we naturally define 0.333...*3 to equal 0.999..., then 1/3 doesn't exist. Likewise, there is no such thing as 1-0.999... .) - Mike Rosoft (talk) 05:37, 25 July 2017 (UTC)

0.(9) ≠ 1[edit]

1) 0.(9) ≠ 1

1.1)
0.9 + 0.1 = 1
0.99 + 0.01 = 1
0.999 + 0.001 = 1
0.99...9 + 0.00...1 = 1
0.(9) + 0.00...1 = 1
0.00...1 ≠ 0 =>
0.(9) ≠ 1

1.2)
0.(9) = 999.../1000...
1 = 999.../999... = 1000.../1000...
999.../1000... ≠ 999.../999... = 1000.../1000... =>
0.(9) ≠ 1


2) 0.(9) < 1

2.1)
0.00...1 > 0 and 0.(9) + 0.00...1 = 1 =>
0.(9) < 1

2.2)
999.../1000... < 999.../999... = 1000.../1000... =>
0.(9) < 1


3) 0.999... -> 1; 999.../1000... -> 1

3.1)
0.9 < 0.99 < 1; 0.99 < 0.999 < 1; 0.999 < 0.9999 < 1; ...
0.99 < 0.999 < 0.9999 =>
0.999... -> 1

3.2)
999.../1000... = 0.999... and 0.999... -> 1 =>
999.../1000… -> 1


4) 0.(9) -> 1

4.1)
0.(9) < 1 and 0.999... -> 1 and 0.999... = 0.(9) =>
0.(9) -> 1

4.2)
999.../1000... < 999.../999... = 1000.../1000... and
and 999.../1000... -> 1 and 999.../1000... = 0.999... = 0.(9) =>
0.(9) -> 1
— Preceding unsigned comment added by 92.101.61.233 (talk) 21:41, 30 August 2017 (UTC)

Until very recently, the article included similar arguments to these showing why . These have now been placed in an appropriate context, so their insufficiency as mathematical proofs is now laid bare. Please refer to the article. Sławomir Biały (talk) 22:28, 30 August 2017 (UTC)

Recovered section heading[edit]

As I see it, the following comments are not inherently created as belonging to the above section, but resulted in this form and layout -without a genuine header- after the complete deletion (Jpgordon) of a discussion (ARB), which had been closed already and contained this meaningless notation "0.00...1", but not the notation alluding to p-adics, contained in the thread above and in Sławomir Biały's comment below. I think it would be advantageous to undelete this one closed discussion, and leave all further da capos of the closed thread as deleted.

Please, treat this edit,especially the new header to your desire. Purgy (talk) 09:15, 20 September 2017 (UTC)
________________________________________________________________________

Yawn. The string "0.00...1" is meaningless in the reals. Thus, this is just a waste of electrons. --jpgordon𝄢𝄆 𝄐𝄇 23:22, 19 September 2017 (UTC)
Not "A waste of electrons". The real number system is a topic in real analysis, not something that the typical reader will have exposure to. The argument
Let , then , so , or
bears a formal similarity to
Let . Then , so , or .
What makes one correct and the other incorrect is the Archimedean property of the real number system. There is nothing about the notation that is inherently meaningless. (Indeed, it is meaningful in p-adic number systems.) An earlier version of the article unfortunately perpetuated the myth that real numbers are defined by decimal notations and certain operations performed on them, and so the identity could then be proven by facile manipulations. The same facile manipulations show just the same that , and probably lots of other equally silly things. The reader isn't served by being fooled into think they understand the reason for the equality, when in fact they do not. The article shouldn't shy away from the defining properties of the real number system. Sławomir Biały (talk) 00:18, 20 September 2017 (UTC)
The ...999 thing strikes me as a rather poor example, given that both the result and the proof are correct, for the 10-adic numbers. --Trovatore (talk) 17:45, 20 September 2017 (UTC)
I think that's why it is a good example. It shows the insufficiency of other "valid" proofs of that also rely on plausible ad hoc rules for manipulating decimal expressions. Nothing has actually been "proven" by either manipulation, unless an interpretation is supplied. Sławomir Biały (talk) 18:40, 20 September 2017 (UTC)
Well, as I pointed out elsewhere, it's not really true that nothing has been proved. What has been proved is that the result holds if the manipulations are valid. Since the manipulations are valid (for the reals in the 0.999... case and for the 10-adics in the ...999 case), the two results do in fact hold. As the manipulations are somewhat believable, this is an incomplete, but nevertheless meaningful, argument to show to learners who do not yet understand the reals in rigorous terms. --Trovatore (talk) 01:15, 21 September 2017 (UTC)
Certainly, we can prove that if the notation satisfies certain axioms, then the sentence is a theorem in that axiomatic system. But "real number" is a specific thing, with a specific set of axioms, and we haven't proved a theorem about real numbers. The axioms of the formal system may be establishable as theorems in the real number system. But those should first be proved. Since their proof is likely to be significantly harder than the supposed proof of the equality of , these algebraic arguments simply beg the question.
Furthermore, the axioms we've settled on for this formal system should not be based on their "believability". My point in bringing up the example is that equally "believable" manipulations lead to other (equally?) strange conclusions. Many of the arguments that regularly appear on this ridiculous "Arguments" subpage are of this kind. Readers shouldn't be encouraged in this way to make up plausible rules for manipulating infinite objects, however true those rules might turn out to be.
Many students, when the meaning of the real number referred to by the notation is actually explained to them, will eventually agree that the thing we just defined is equal to one. But students usually do not have a clear idea of what is meant by this notation in the first place, so it is pointless to attempt to "prove" that something they don't understand is equal to something else on the basis of plausible-seeming rules. Worse, students will often think they understand what is, but don't. Sławomir Biały (talk) 02:04, 21 September 2017 (UTC)
@Antonboat: I am sorry to say that several well-versed mathematicians have spent far more time than is appropriate on this subject. Ultimately, it is not going to be productive for us, or you, if you resist or reject our advice for gaining a better understanding of this.--Jasper Deng (talk) 09:13, 9 January 2018 (UTC)
The following discussion has been closed. Please do not modify it.

Comments of/additions to?, Formal proof[edit]

The in "Formal proof" given additon rule for decimal numbers implies

But than

and if

than must equal 0. In that case either 1 or must be equal to 0. But we know that 1 as well as is > 0. Ergo: must be for every and thus .

Euclid's reductio ad absurdum proved that assuming a possibly largest prime number was an absurdity, <ref> https://en.wikipedia.org/wiki/Largest_known_prime_number<ref>.
In “Formal proof” is stated:
This proof relies on the fact that zero is the only nonnegative number that is less than all inverses of integers, or equivalently that there is no number that is larger than every integer.

But since all positive prime numbers are odd numbers, it follows from Euclid’s proof that the assumption of a possibly largest positive integer , either even or odd, must be absurd too. So for every there must be a number , being a number ≥ 1, that is (much) larger than . From this follows that < > 0. ˜˜˜˜

You're assuming that if something s true for every natural number it is true at the limit. This is like saying all natural numbers are finite, therefore infinity does not exist. The correct conclusion is we can only talk about the finite numbers unless we have an extra axiom.. In this case we can't say anything about the limit without the Archimedean property or completeness axiom as discussed near the start of the article. Dmcq (talk) 12:35, 2 December 2017 (UTC)
Dear madam/sir, Thank you for your comment. You write: “You're assuming that if something s true for every natural number it is true at the limit.” I’m sorry, but no, I’m not asssuming that. Using Euclid’s Reductio I contend that it is impossible that there could be a limit to the infinity of possible natural numbers. In my opinion logically, I conclude that for every natural number one can create or think of, there must be a number , ≥ 1. Where do you think lies the error in my logic. Can you specify a limit with regard to the infinite multitude of natural numbers? Or can you prove that Euclid ‘s Reductio is not 'true'? Antonboat (talk) 11:56, 3 December 2017 (UTC)
It is perfectly true that 0.999...999 with any natural number of nines denotes a real number that is less than 1. But "0.999..." is shorthand for saying that there are an infinite number of nines. Infinity is not a natural number. So while you have indeed proved that 0.9, 0.99, 0.999, and so on are all less than 1 with any arbitrary natural number of nines, it does not follow that 0.999..., which has an infinite number of nines, is less than 1. In fact it is not.
If you are just uncomfortable with the idea of an infinity that is larger than any of the naturals, there is another approach. 0.999... can be defined as the limit of that sequence (0.9, 0.99, 0.999, ...), i.e. a number separate from the sequence which these terms get as close as you want to (but might not reach). We can also see that as you go further and further down that sequence you will hit terms that are as close to 1 as you like: if you would like a term that is no further than 10n from 1, take the (n+1)st term. And the sequence doesn't fool around: once the terms get within this distance of 1, they don't move back away, and as you get further and further in the sequence the distance to 1 gets below any nonzero real number you want, without reaching zero, of course. So the limit is equal to 1.
We also defined 0.999... to mean this limit, so 0.999... = 1. Again, the limit is not a member of the sequence, so just because you have correctly proved that every member of the sequence is less than 1 does not preclude its limit from being 1. Double sharp (talk) 12:52, 3 December 2017 (UTC)
Dear Double sharp. You write: “ But "" is shorthand for saying that there are an infinite number of nines. Infinity is not a natural number.” Euclid showed us that it is impossible to express the infinite multitude of nines in a natural number because there is no limit to this multitude. Please send me an concrete example of the transition from a natural number to a non-natural infinite number.
You write: “If you are just uncomfortable with the idea of an infinity that is larger than any of the naturals, there is another approach. can be defined as the limit of that sequence (), i.e. a number separate from the sequence which these terms get as close as you want to (but might not reach).” I do not agree with you, your: “(but might not reach)”, must, in my opinion because of Euclid’s Reductio, read: ”(but shall never reach)” And because of this in my opinion must be interpreted as an infinite sequence of real numbers .Antonboat (talk) 16:24, 4 December 2017 (UTC)
Some sequences reach their limit. (1, 1, 1, 1, ...) certainly does, for example. It's just that this one does not.
Again, 0.999... doesn't represent the sequence: it represents the limit of the sequence. It's not part of the sequence (0.9, 0.99, 0.999, ...): it's specifically defined to be the number that these terms approach, which they happen not to reach. So saying that 0.999... = 1 doesn't mean that we have to have some number with a finite number of nines that is equal to 1. Double sharp (talk) 06:05, 10 December 2017 (UTC)
You seem to be asking for a proof of the existence of an actual infinity. We can't do that - it all depends on having an extra axiom as I said before. But I can indicate the existence with a simple example. For every natural number N we can associate an apple with each number up to N - but we can't redistribute the apples so there is no apple associated with one of the numbers and all the other numbers have one apple at most associated. However with all the natural numbers we could redistribute the apples so the apple that was associated with number n is now associated with 2n. This means there is now just one apple associated with each even number and none with the odd numbers. This means that even though something is true for every particular N it is not true for all the numbers. This is an example of N going to the limit and something not then being true. Dmcq (talk) 13:04, 3 December 2017 (UTC)
Dear Dmcq. You write: “You seem to be asking for a proof of the existence of an actual infinity. We can't do that - it all depends on having an extra axiom as I said before.” And indeed I am asking for such a proof. And I knew beforehand that you could not produce that. Because Euclid’s Reductio showed us all that the assumption of an actual infinity concerning what ever, be it numbers, time or lines is an absurdity. That’s why infinity cannot be a number. Therefore an axiom: “there is an actual infinity”, must be contrary to Euclid’s Reductio.
You conclude your example: “This means that even though something is true for every particular N it is not true for all the numbers. This is an example of N going to the limit and something not then being true.” But concerning the whole numbers Euclid’s Reductio implies that the infinite multiplicity of natural numbers has no limit, therefore one can not speak of "all numbers", unless ... one starts from an (according to Euclid’s Reductio) absurd axiom: “There is an actual infinity”.Antonboat (talk) 16:24, 4 December 2017 (UTC)
Yes, Antonboat, indeed, as you say, "every number in the infinte sequence is smaller than 1", but we can prove that we can "get as close to 1 as we like", in the following sense: you give me any positive distance you want, and I can give you a number in the sequence that is closer to 1 than your chosen distance. (Go ahead, try it, give me a distance of your choice.) That can be proven, and that is abbreviated as . It is that simple, and the article says it at the end of section 0.999...#Infinite series and sequences. That is the only section in the entire article that matters. You can safely forget everything else. - DVdm (talk) 16:45, 4 December 2017 (UTC)Antonboat (talk) 18:56, 4 December 2017 (UTC)
Dear Dvdm. I am sure that you can give me a number in the sequence that is closer to 1 for every distance I should choose. Because, as I wrote in my original note, this follows from < > , being ≥ 1, for any one has chosen. But can you give me a real number in the aforementioned sequence for which = 1?
No, of course not. There would be no reason to give such a number. - DVdm (talk) 19:13, 4 December 2017 (UTC)
Just saying Reductio doesn't show anything if you don't actually cover the various options. And you haven't. Without some extra axiom .99999... does not have a particular value. With the Archimedean axiom it is 1. There are other systems in which infinitely small numbers exist so it is infinitesimally less than 1, but I get the feeling you'd find that even less palatable, the standard number system does not have numbers which are different but cannot be distinguished by a sufficiently good ruler. Dmcq (talk) 17:22, 4 December 2017 (UTC)
Dear Dmcq. You’re right as you say: “Just saying Reductio doesn't show anything if you don't actually cover the various options. And you haven't.“ In the future I’ll try not to do it again. But please correct me in that respect if I fail and my argument becomes unclear. You wrote: “Without some extra axiom does not have a particular value.” I agree. I try to gain more insight into the mathematical view of the infinitely small and (actual) infinite. In that context, I consider the assumption of infinitesimals > 0 as not in conflict with the expression: < 1 and with Euclid's in the original note extensively explained Reductio.Antonboat (talk) 18:56, 4 December 2017 (UTC)
Dear Dmcq. You write: “Without some extra axiom does not have a particular value. With the Archimedean axiom it is 1.” In: https://en.wikipedia.org/wiki/Axiom is said: “As used in mathematics ... an axiom is any mathematical statement that serves as a starting point from which other statements are logically derived.“ Starting from the Archimedean Axiom you argue that . Starting from Euclids Reductio I argue that . Could it be possible that we both be right?Antonboat (talk) 14:39, 5 December 2017 (UTC)
Just because you think something is absurd does not mean that it is absurd or wrong. Reductio ad absurdum is simply the principle that a supposition that proves a falsity must be wrong, and it predates Euclid. You have not logically proved anything at all. The article covers the various options. In the standard model of the real numbers the value is 1. In some other systems infinitesimals exist. The axioms are different for the different systems. See the section 'In alternative number systems' in the article. Dmcq (talk) 18:22, 5 December 2017 (UTC)
Dear Dmcq,The reason I did not respond at once to your comment is that I did not get a signal that there were any responses. Reading your comment I realize that instead of writing “Reductio ad Absurdum” I better should have written: “Euclid's Theorem” https://en.wikipedia.org/wiki/Euclid%27s_theorem. You write: “Just because you think something is absurd does not mean that it is absurd or wrong.”
First a linguistic aside. As my native language is Dutch we say “absurd” or: “ongerijmd”. For this last word Google translate advises the translation: ¨absurd”. In Dutch the word “ongerijmd” means :“te onwaarschijnlijk om waar te kunnen zijn”, in translation: “too unlikely to be true” . I think this fits best with the word “illogical”: according to Google meaning: “lacking sense or clear, sound reasoning.”
I wrote that assuming an actual infinity, speaking of: all the real numbers, etc is absurd. (But from now on I‘l write “illogical” instead of “absurd”.) I wrote this on the authority of Euclid’s proof. This proof convinced me that neither a largest prime number, nor a largest natural number can exist. So, once again, see: 11:56, 3 December 2017, can you prove that Euclid ‘s Theorem is not 'true'?Antonboat (talk) 10:42, 11 December 2017 (UTC)
Take the decimal expansion 0.999..., with one 9 for each decimal place. Now we can index those decimal places as {1, 2, 3, ...}, corresponding to the natural numbers: one 9 for each natural number. But in that case no natural number suffices to represent how many 9's there are, because there will still be more places behind corresponding to the natural numbers after it. From what I see from below, you appear reluctant to accept an actual infinity, because given any natural number there is always one after it. But that does not exclude the possibility of an actual infinity that is not a natural number.
The bottom line is that your reductio ad absurdum relies on properties of the natural numbers, while 0.999... does not have a natural number of 9's. Thus your argument is powerless to say anything about it. All it can say is that every member of the sequence {0.9, 0.99, 0.999, ...} is less than 1, but 0.999... is not in that sequence, because the elements of that sequence can be indexed with the naturals {1, 2, 3, ...} by how many 9's there are in their expansion, and 0.999... doesn't have a natural number of 9's. You need an extra axiom indeed to say anything about it. But it is not the case that your argument proves that 0.999... < 1. Rather, starting from the Archimedean axiom you get 0.999... = 1; without it, 0.999... and its properties stand outside your system. And something like this is indeed covered by the article under #Ultrafinitism: under that view, 0.999... is considered to be meaningless because of the complete rejection of all infinities. But you will find it very hard to get much stuff done that way, which is why it does not have widespread acceptance. Double sharp (talk) 06:05, 10 December 2017 (UTC)
Dear Doublesharp. You write: “You appear reluctant to accept an actual infinity, because given any natural number there is always one after it. But that does not exclude the possibility of an actual infinity that is not a natural number. ... The bottom line is that your reductio ad absurdum relies on properties of the natural numbers, while 0.999... does not have a natural number of 9's.”
I’m sorry, but I can’t agree with you. Euclid’s theorem https://en.wikipedia.org/wiki/Euclid%27s_theorem proofs by a reductio ab absurdum that the assumption of a largest prime is illogical. As every prime is an uneven natural number and an even natural number is 1 larger than it’s preceding uneven number, the assumption of a largest natural number must illogical too. And when there is no largest natural number the multitude of natural numbers must be infinite. As infinite as the number of nines in 0.999…. Therefore, this reasoning leaves no room for a sequence of natural numbers that ceases as such and "somewhere, but can you tell me where exactly?" turns into a sequence of infinite whole? numbers.Antonboat (talk) 12:23, 11 December 2017 (UTC)
I'm not saying there is a largest natural number. I'm also not saying that the sequence of natural numbers stops somewhere. And indeed that sequence is infinite. What I am saying is that infinity is not a natural number. It is separate from them, as if you were counting "1, 2, 3, ...", and after all the natural numbers came infinity, like ω in the ordinals.
If you like, you can imagine counting "1" at t = 0.9 s, "2" at 0.99 s, and so on, and after 1 second you've counted every natural number at some point. Then after that you can deal with infinities, but you have to keep in mind that they are not natural numbers, so that properties that are true for natural numbers may not apply there. The number of 9's in 0.999... is not a natural number. So your "proof", which relies on properties of natural numbers, does not apply to 0.999..., although it does apply to 0.9, 0.99, 0.999, and so on. Does this clarify things? Double sharp (talk) 08:42, 13 December 2017 (UTC)
Dear Doublesharp. In answer to your question: “Does this clarify things?”, the following answer. You are crystal clear in your position concerning the relation naturel versus infinite numbers as you write: “What I am saying is that infinity is not a natural number.“ You are right because in my opinion infinity cannot be a number, neither natural number, nor a real number, nor an ordinal number Aleph, nor whatever other kind of number someone could assign to infinity. (But please do not take this statement very seriously. The person who writes this is a non-mathematician. And who is inclined to take a psychologist seriously when he makes mathematical statements, but not proofs them?) You write that the natural numbers are separate from the natural numbers. But I think you’ll agree with me that infinite numbers, if they could be a logical and mathematical possibility, should be > 1. And if that is true and if an infinite number is no natural number, than the infinite numbers must, according to my logic, be a follow up of the natural numbers, as in: 1, 2, 3, ..., Aleph?/ω?. N.B. If we assign an ordinal to each element of this sequence as follows: 1, 2, 3, ..., Aleph?/ω?, logically the sequence of ordinals and the sequence of naturals + infinite numbers must be identical to each other. Is this proof enough?Antonboat (talk) 14:26, 13 December 2017 (UTC)
Yes, that's pretty much how the ordinals work, extending the naturals by coming after all of them: first we have 0 as an equivalence class as (just read that as "corresponding to", if you like) the set {}, 1 as that of {0}, 2 as that of {0, 1}, and so forth. Then ω comes after all the naturals as the equivalence class of {0, 1, 2, 3, ...} = N0, and then we have ω+1 corresponding to {0, 1, 2, 3, ... ω}, and then ω+2, ω+3, ..., ω+ω=ω2, ..., ω3, ..., ωω = ω2, and then ω3 and all the way to ωω, ωωω, and so forth. Then we get things like ε0, defined as the smallest solution of the equation ωx = x, and they keep going. But I don't see why you think that they can't be numbers: it really depends on what you want a number to be. Already once you get past the naturals you are generalising things, because they no longer count anything discrete. So we are taking some familiar properties of the naturals, generalising them into a bigger structure, trying to preserve some properties and gain some interesting ones, though of course we cannot keep all of them. Put like that, I don't see why ω shouldn't be a "number", if a specific kind of number.
But it's also important to note that these are not natural numbers. That's why they don't contradict the obvious argument that you can't have a largest natural number because if there was one, call it N, then N+1 is bigger. This argument is valid and true, but it doesn't rule out the possibility that you can have something that is bigger than all the natural numbers. In some systems you can even have it as the biggest object, like the extended real number line, and it works out because in that system, ∞ + 1 = ∞, so the argument doesn't hold because ∞ + 1 isn't greater than ∞. And no amount of adding together natural numbers will get you to ∞, which violates the Archimedean property (that any small number, when added to itself enough times, will surpass any larger number) that the reals have.
The reals are Archimedean: thus they don't have infinite numbers and they don't have infinitesimal numbers. (I do need to clarify that 0.999... is a real: it has an infinite number of 9's in its decimal expansion, but it isn't infinite itself: for one, it's obviously less than 2. That is all right.) So we can immediately see that 0.999... = 1 by computing 1 − 0.999... and seeing that it must be less than any positive number. (For instance, it's less than 0.0001 because 0.999999... > 0.9999 and 1 − 0.9999 = 0.0001, and you can make this argument by truncating the expansion 0.999... after any number n of digits and see that it's less than 10n.) It also obviously can't be negative because 1 can't be less than 0.999... (I trust this bit is obvious). But once you say that, the difference 1 − 0.999... has to be zero, because the reals have no infinitesimals and so the only real number that's nonnegative and yet less than all the positive numbers is zero. Again, there are other systems, but the article is about 0.999... in the reals, and there it equals 1. This could be considered a corrected version of your proof at the start of this section, if you like: it takes your correct observation for decimals with finite numbers of 9's after the point, and then makes the leap to the categorically different case where there are an infinite number of 9's. Double sharp (talk) 14:49, 13 December 2017 (UTC)
Dear Double Sharp. Thank you for your many comments. It seems you comments faster than I can react. You write: ... “I don't see why ω shouldn't be a "number", if a specific kind of number. ... But it's also important to note that these are not natural numbers. “ Writing about infinite numbers you say: “But it's also important to note that these are not natural numbers. That's why they don't contradict the obvious argument that you can't have a largest natural number because if there was one, call it N, then N+1 is bigger.” We can easely see whether a number is < 1 or > 1. But how can you see which is a natural number and which is not? The multitude of natural numbers is infinite in the sense that for whatever value of the natural n we can create a number n + 1 > n.
I’m interested too in the way how to reach 1 from any 10E-n which is below any nonzero real number. You wrote (12:52, 3 December 2017 (UTC): ... “as you get further and further in the sequence (you meant: 0.999...) the distance to 1 gets below any nonzero real number you want, without reaching zero, of course.)" And you say: “the only real number that's nonnegative and yet less than all the positive numbers is zero.” I know that 0 is a digit. But how can a number be zero? By an axiom: 0 is a number? And how can you travel a distance, making steps zero units long? (NB It may be that I did not understand what you wrote, or that I did not correctly take the quotations from their context.)Antonboat (talk) 21:27, 14 December 2017 (UTC)
Dear Double sharp. Please forget the comment above because I think it is not clear enough. My intention, corresponding with you, which I appreciate very much, is:
a) to find out how mathematicians think they can succeed in achieving 0.(9) = 1, in smaller steps than the reals in the decreasing sequence: 0.1. 0.01, 0.001, ... and
b) in what way you came to accept 'infinity' as a number, and
c) in what way we can reach this number, if the naturals do not offer this, in concreto: starting from 1 , 2, 3, .... until the number 'infinity'.Antonboat (talk) 09:02, 15 December 2017 (UTC)
I really am not sure what you mean by (a). If the idea is that the numbers {0.9, 0.99, 0.999, ...} are respectively distance {0.1, 0.01, 0.001, ...} from 1, meaning that if you keep adding nines one at a time to the decimal you get closer and closer but never reach 1, that is all true. No matter how far you go in the sequence, if you stop anywhere, you don't reach 1. But "0.999..." with an infinite number of 9's means that you never stop. As for (b) and (c), before I can answer this effectively, I'd need to know: what exactly do you consider to be legitimate numbers? Evidently you don't seem to accept that 0 is a number. Do you think 1/2 is a number? −1? √2? e? i? j?
As for 0 itself: why not have it as a number? It certainly behaves consistently when you add it to the other natural numbers, just defining it as the result of 1 − 1. You can certainly do addition, subtraction, and multiplication with it, and you can even do division as long as it's the numerator and not the denominator. And in fact, the very place-value notation implies the acceptance of zero as a number that you can use as an addend: just as 583 means 5 × 102 + 8 × 101 + 3 × 100 = 500 + 80 + 3, so 503 means 5 × 102 + 0 × 101 + 3 × 100 = 500 + 0 + 3. Honestly, this sort of question feels like how the Greeks rejected 1 as a number, because a number to them had to be a multitude. Mathematicians tend not to get themselves involved in such philosophical quibbles. Rather we ask for some property we want for our original structure, and obtain it by extending that structure in some way. That is how we get from N to Z: we want a ring that we can embed the naturals in (in layman's terms, we want a structure in which subtraction is always possible). And similarly it is how we get from Z to Q: we want a field that we can embed the rationals in (in layman's terms, we want a structure in which division by anything but zero is always possible). And it helps that there are very many natural situations modelled by Z and Q that practically compel us to accept their legitimacy as structures, even as number systems, if we want to get pretty much anything done.
Now once we've accepted that, I'm having difficulty understanding what the difficulty is. Sure, you can never get to infinity by counting up with the naturals. But then again, you also can never get to 0.5 by counting up with the naturals. And why should that matter? Neither are natural numbers. The naturals don't give us everything we want, so we extended them into larger systems that do. Double sharp (talk) 16:05, 16 December 2017 (UTC)
Dear Double sharp. I knew that I was not clear enough in my comments to you. I've spent the weekend to get things more clear. This has resulted in e new section: “Why 0.999... = 1 cannot be true and infinite cannot be neither a natural, nor an infinite number.”Antonboat (talk) 21:03, 18 December 2017 (UTC)

A real larger than any real in the infinite representation of (9) 0.(9)[edit]

It is said in: https://en.wikipedia.org/wiki/0.999... “In mathematics, 0.999... (also written 0.9, among other ways), denotes the repeating decimal consisting of infinitely many 9 after the decimal point (and one 0 before it). This repeating decimal represents the smallest number no less than all decimal numbers 0.9, 0.99, 0.999, etc. “ In this comment I write instead of: . Following what is said above about the infinite sequence: must be represented by . The first number in the representation of ( is larger: than the first real in the representation of : . So in the representation of must be a real larger than any real in the representation of .Antonboat (talk) 09:16, 11 December 2017 (UTC)Antonboat (talk) 19:10, 18 December 2017 (UTC)

I think using this notation, which avoids the explicit ellipses, seduces you to introduce a concept, not covered by the usual setting of decimals. The notation 0.999..., as said, denotes a "repeating decimal", and is not prepared to be amended to 0.999...9, because this latter notation totally loses the meaning of a "repeating decimal". It denotes a decimal with an unspecified, but finite number of 9s right to the decimal point. All this mixing up of different notations does not lead to any meaningful result. Similar holds for the somehow confused notation above of 999...9. If you saw ...999 before, this belongs to the number system of p-adic numbers, with quite different properties compared to the usual numbers, and also does not allow for 9...999.
Please, allow me to suggest that, in case you are interested in continuing beyond or , you should consult ordinal number or cardinal number, but stop tinkering with decimals. They are not apt for this task, because they are just a notational convention, no good selection for contemplating the concept of number.

And finally: sorry, but I believe you are totally wrong at this place. I think, you should consult the Help desk, or similar. Someone might close this thread, or even worse, try to delete this whole subpage, again. Purgy (talk) 10:38, 11 December 2017 (UTC)

Dear Purgy. Thank you for your suggestion. I’m a Dutch psychologist, very interested in what mathematicians say and think about infinity. And I’m more than very interested in an concrete example of the transition from a natural number to a non-natural infinite number as Aleph, as nice and simple as the way in which Euclid proofs that the assumption of a largest prime is illogical. Can you send me that? I would be very grateful to you.
As for your second comment: my note is intended as a constructive contribution to the debate about 9999 .... I expect to receive constructive and enlightening reactions like yours. And such a reaction could be that people prove to me "claire et distincte" that what I argue is pure nonsense.Antonboat (talk) 11:38, 11 December 2017 (UTC)
Dear Purgy. As an afterthought another comment. You write: “I think using this notation, (you ment , Antonboat), .... , seduces you to introduce a concept, not covered by the usual setting of decimals." You’re right. But your interpretation of  : does not match with what I had in mind when I created this notation. I came up with that idea when I read in: Wikipedia "0.999 ..." that is represented by the sequence: . Analogous to that, I created the notation representing the sequence: . By means of the notation I wanted to indicate a repeating nine to the right of the decimal point, immediately followed by a nine, like this: . It turned out that the notation could be of excellent use to prove that . The proof goes as follows:
. (Analogous to: ).
. (Analogous to: ).
.
.
Ergo: , and thus , must be < .
As you write: “All this mixing up of different notations does not lead to any meaningful result.”, I do not believe that you will be very interested in my argument above. But given your fair response on my note, I thought I owed you this explanation. It seems to me very useful in a proof as above mentioned and as example of a way to multiply notations as . And reading the wide variation in notation and definition in what is written about on Wikipedia, in my opinion, the notation is in this variety not out of place.Antonboat (talk) 12:50, 13 December 2017 (UTC)
Again, I must repeat: when mathematicians write 0.999... and expect the standard meaning, they don't mean the sequence {0.9, 0.99, 0.999, ...}; they mean its least upper bound.
And once again you're tacitly assuming that there's a specific finite number of digits enclosed in the parenthesis. For example, it is true that if you expand out all those parentheses into four digits, you get 9 * 0.99999 = 8.99991, then 9 − 8.99991 = 0.00009, 0.00009 / 9 = 0.00001, and then 0.99999 is obviously 0.00001 away from 1. But here's the thing: 0.999... has more than four 9's there. Well, then, put in ten nines instead, repeat those steps, end up with 0.00000000001, and 0.9999999999 is obviously 0.0000000001 (10−10) away from 1. But again, 0.999... has more than ten 9's there. No matter how many 9's you put in, 0.999... has more 9's than that. If you put in n 9's, you get 10n as your distance from 1. But again, 0.999... has more than n 9's, and so it's closer than 10n, no matter what n you put there. Which means that the distance of 0.999... from 1 has to be 0 in the reals, as that's the only nonnegative number smaller than every negative power of ten. Does that make sense? Double sharp (talk) 14:57, 13 December 2017 (UTC)
Dear Double sharp. I have read both your comments of 13 December and my answers to some of it it may be found in the new section “Why 0.999... = 1 cannot be true and infinite cannot be neither a natural, nor an infinite number.”Antonboat (talk) 21:17, 18 December 2017 (UTC)
In the reals there are no infinities, and any infinities you see are not real numbers and don't play by their rules. In particular, we can only deal with finite negative powers of ten. You can have a 10^(-1) place, a 10^(-2) place, and so on, but there is no 10^(-ω) place there. There is only one for 10^(-n) for each natural number n, which is enough to get you an infinite number of decimal places, but not enough to continue before that. Therefore, in the reals, trying to have an infinite repeating sequence of 9's and then something else is indeed nonsense. It is indeed not nonsense in some different systems, such as the hyperreal numbers. But: (1) this article is covering the reals, which are the most generally used system, and there 0.999... = 1; and (2) the hyperreal with nines all the way even past infinity, i.e. 0.999...;...999..., still equals 1. Double sharp (talk) 09:01, 13 December 2017 (UTC)

This whole discussion highlights the problematic nature of symbol-pushing arguments such as those that, until recently, were presented as evidence of the equality in the article. Such arguments based on the manipulation of infinite decimal expansions are inherently problematic, because the mathematical notion of infinity is rather tricky. Indeed, there are many paradoxes of infinity that show that dealing with infinite objects requires extreme care (e.g., Hilbert's paradox of the Grand Hotel). In any case, one is certainly free to define the notation to refer to many different things such as an infinite expression which is manifestly not the same as the expression . A typical way to define the real number referred to by an infinite decimal expansion is the least upper bound of its finite decimal truncations. Thus is, by definition, the least upper bound of the sequence . This least upper bound is equal to one, not because of symbol-pushing arguments, but rather because of the Archimedean property of the real number system. Sławomir Biały (talk) 14:40, 13 December 2017 (UTC)

Dear Slawomir Bialy Thank you for your comment. You write: “This least upper bound is equal to one, not because of symbol-pushing arguments, but rather because of the Archimedean property of the real number system.” So it seems I am on the wrong trail with this note.Antonboat (talk) 14:56, 14 December 2017 (UTC)
I wouldn't say "wrong track". What it illustrates though is just how problematic infinite objects can be, which is part of the point of the exercise. There was a huge discussion recently about a section of the article where symbol-pushing arguments were presented as evidence of the equality, with necessary and important details left out. I and a few others were of the opinion that such arguments should not be convincing to anyone that actually needs to be convinced, because such arguments can be readily modified to prove all kinds of silliness, like . Sławomir Biały (talk) 15:17, 14 December 2017 (UTC)

@Antonboat, honestly, I want to get out of this treadmill. Double sharp seems to refer to "your" idea of ω, hidden behind that unlucky 0.(9)9, where I coined the 0.(9) as "not prepared to be amended" in the context of a "repeating decimals". The quintessence is to me that you go beyond the setting for this 0.(9). OTOH, Sławomir Biały reminds you of the correct use of "sequence ", which does not cover your interpretation of "representing 0.(9)", but is just the sequence , which neither contains 1 nor 0.(9), but converges within the reals to 1, giving rise to the definition of the meaning for these awkward notations of real numbers 0.(9) or 0,999..., both equaling to 1.

Sorry, I cannot help you any further, but to exhort you to be more strict and coherent in using mathematical notation: There is yet absolutely no justification for abusing the standard routine for multiplying decimals of finite length for your non-decimals. "It works" is no valid argument. One last thought as good bye: 1 is the only number which has infinitely many such sequence elements in any ɛ-neighbourhood. Purgy (talk) 18:06, 13 December 2017 (UTC)

Dear Purgy. You write: “Double sharp seems to refer to "your" idea of ω, ...“, but that is not true. I used the symbol for omega in his comment, in my comment to him. You write: “There is yet absolutely no justification for abusing the standard routine for multiplying decimals of finite length for your non-decimals.” I’ll keep it in mind and won’t do it again. I now consider this note as a dead end.
You write ”Antonboat, honestly, I want to get out of this treadmill” The "mores" of Wikipedia assume that we Wikipedians be polite and constructive in communicating on Wikipedia. But we’re not obliged to respond if we do not want it or can not do that for any reason whatsoever. Thank you for your comments and: “Alle goeds “. That’s Dutch for “All the best”. — Preceding unsigned comment added by Antonboat (talkcontribs) 14:19, 14 December 2017 (UTC) Antonboat (talk) 14:52, 14 December 2017 (UTC)
So you have 0.(9)9 meaning an infinite sequence of 9's and then a 9! That reminds me of

  π goes on and on
     And e is just as cursed
     I wonder, how does π begin
     When its digits are reversed?

by Martin Gardner. Dmcq (talk) 00:15, 15 December 2017 (UTC)
Dear Dmcq. Thank you for your funny respond to my idea of 0.(9)9. There is the same logic in that idea as expressed in the little poem you sent me. The notion 0.(9)9 emphasizes that we can not record the number of nines by means of the note 0.(9) because the repeating nine exceeds every representation that we can hink of. That’s why in my opinion 0.(9)9 = 0. (9). The poem states: “π goes on and on.” In the same way the nines in 0.(9) and in 0.(9)9 go on and on. “When its digits are reversed” we find for 0.(9) a natural number (9).0 that goes on and on because the smallest natural number with a nine is 9, but starting from Euclid’s Theorem, there cannot be a largest natural number consisting of only nines, nor a limit to it as the (non?)natural number Aleph, Omega or Infinite. Communicating with you and with all the Wikipedians who showed me the honor (really, I feel it that way) to respond to my notes, I noticed that you are all very much convinced of the possibility of such an infinite number. Following Euclid I do not see that possibility. To be clear: my intention is not to convert you to my beliefs. (It would be impossible I know.) I am not a missionary. My intention, corresponding with you is,
a) to find out how mathematicians think they can succeed in achieving 0.(9) = 1, in smaller steps than the reals in the decreasing sequence: 0.1. 0.01, 0.001, ... and
b) in what way you came to accept 'infinity' as a number, and
c) in what way we can reach this number, if the naturals do not offer this, in concreto: starting from 1 , 2, 3, .... until the number 'infinity'.Antonboat (talk) 08:56, 15 December 2017 (UTC)
@Antonboat: It's hard for me to follow what you're arguing, as I'm unable to see it as more than hand-waving. As Purgy said above, we are unable to have this conversation unless you are willing to propose a rigorous argument - and per WP:NOTAFORUM we shouldn't even talk about that, unless it is directly related to improving the article.--Jasper Deng (talk) 09:16, 15 December 2017 (UTC)
Dear Jasper deng. I am a native Dutchman, so I speak and write Dutch better than you do, but my English could be therefore (very) imperfect. I was not familiar with the expression: "handwaving." Searching on Wikipedia (thank you for your link) I found out that it means in Dutch something as: “uit je nek kletsen”, translating in English: ≈ “to chatter from your neck”. Well that’s not a friendly evaluation of my contributions to Wikipedia, but it’s an opinion. Thank you for that. I use this argument page tot confront Euclids theorem with what is said in Wikipedia “999...”. So what you see as my handwaving: in slang Dutch: “ Mijn gelul” is for me “arguing” and a serious preoccupation. Antonboat (talk) 10:48, 15 December 2017 (UTC)
Dear Jasper deng. I've started a new section: “Why 0.999... = 1 cannot be true and infinite cannot be neither a natural, nor an infinite number.” The beginning of it is as follows: "This note is a response to Jasper deng who said that my notation 0.(9)9 and what I said about it was ‘hand-waving’.Dear Jasper deng. Let me show you by simple logical reasoning that you’re very wrong in that." I invite you to react on this section.Antonboat (talk) 20:52, 18 December 2017 (UTC)
Jasper, this is the arguments page. It's specifically for people to make arguments that don't have any immediate application to the article. --Trovatore (talk) 09:25, 15 December 2017 (UTC)
Right, but WP:NOTAFORUM is a project-wide policy and at the least, it sets a limit on discussions like this one.--Jasper Deng (talk) 09:56, 15 December 2017 (UTC)
Like most of WP:NOT, NOTAFORUM is mostly about article space. (It fairly explicitly excludes the refdesks, for example.) A lot of NOT would be seriously constraining in other spaces. The limits in article-talk space come mainly from WP:TPG, which the arguments pages are specifically designed to relax. --Trovatore (talk) 10:26, 15 December 2017 (UTC)

Three questions on 0.999...[edit]

Primarily, I do not want to appear as impolite and unhelpful. It is just that I am constantly afraid of appearing such, because of my awkward use of the English language as a non-native, not even living in an English speaking country. Secondly, it is true that I want, for egoistic reasons, to reduce my involvement in this discussion, but as far as possibly without any hard feelings.

Under this preamble I want to restate my positions, which are strongly constricted to to aspects within "real numbers".

  • ad a): In my perspective, the use of 0.(9) is already "hand-waving" to me (please, under all circumstances, ignore the "pejorative"-qualification, and skip to the "academic use" in the linked article!) within the reals. In my opinion, decimal representations of reals are unwieldy to explore the boundaries of theoretical properties of the reals. There is not even a bijection (proven!) between reals and decimals. Decimals gain their importance for the ubiquitous representation of real world effects, fictitiously measured within the concept of real numbers (they are rational, at best!). Any use of symbols, hinting to "infinity", like ellipsis, or (.), or ∞, ... requires a strictly defined background to allow further manipulation. All these backgrounds employ (to my knowledge) the concept of "limits", hard enough to define on the basis of abstract numbers, avoiding as much as possible their representation as decimals (free monoid, made up by the star of the alphabet of the 10 digits?). Once a specific limit-expression, accepted as definiens of the definiendum 0.(9), is found, a rigorous mathematical treatment may start. Please, note that this acceptance is via "defining a mathematical meaning for 0.(9) (= 0.999...) := blablabla" (I don't know why WP deprecates the ":="). Nobody is allowed to modify this definition within the system under discussion, without proving the equivalence, or explicitly setting up a new notion. It is simply not the case that 0.(9)9 (= 0.999...9) := blablabla9". Obviously, 0.(9)9 requires a new definition, since it violates the inaccessibility of the end of an "infinite chain" in the system of real numbers. Alluding to the poem: there is no last digit of π to start πR. This is not to say that one cannot construct "numbers", which go beyond this "infinity in length" of decimals, there is, afaik, a rich hierarchy of "infinities" and unclear conjectures about their connection to the "continuum". I strongly believe, however, that there are only a few mathematicians (Wildberger?), who already deny the "infinity in length", negating the existence of reals, and the other, so I assume, do not strive for "succeeding to achieve 0.(9) = 1 in smaller steps", because, firstly, this is an equality-fullstop (according to the definition of 0.(9)!), nothing to achieve in steps, and second, there -proven!- are no smaller steps within the reals as the reals already offer (tautology?). You must change your axioms, if you want to go there (infinitesimals). BTW, they are not too handy and not widespread.
  • ad b) Personally, I do not accept "infinity" as a number. As said, within the reals it is an abbreviation to me, in settings that explicitly need "limits". Sometimes the limits get so trivial that one infinity (∞), or both (±∞) are adjoined to settings of numbers, and so are treated as numbers, leading to abuse of notation like -1/0 = -∞, or 1/∞ = 0, (0, ∞). In the context of the "higher infinities" there are additional rules for calculation, and so I sometimes ask myself if these objects are rightfully coined "numbers". Possibly, the whole use of the word "infinity" in math is a revenge of philosophy for math having gotten grown up, out of this ancient guardianship: no more mathematicians to be drowned by philosophers for small irrationalities. :)
  • ad c) I simply do not know, if "to be reached" is a defined property for these numbers. So, sorry, this question is "hand-waving". However, I deny "infinity" to be a number in this context. :p

Please, do not assume any hostility from my side if I stopped commenting with this. “Alle goeds!“ Purgy (talk) 16:16, 15 December 2017 (UTC)

Why 0.999... = 1 cannot be true and infinite can neither be a natural, nor an infinite number.[edit]

This note is a response to Jasper deng who said that my notation 0.(9)9 in: “A real larger than any real in the infinite representation 0.(9) “ and what I said about it was ‘hand-waving’. Dear Jasper deng. Let me show you by simple logical reasoning that you’re very wrong in that. I suspected that it would be clear to the mathematicians who read the expression 0.(9)9 in my above mentioned note that they would understand that it was a provocation, because I meant it that way. But because some reacted very shocked on it, I let it go. It was clear to me than, that I had to look for a better way to express my interpretation of 'infinity'. Let me show you that the expression 0.(9)9 in my opinion is not hand-waving. But first I’ll show you that the notation 0.999... = 1 is conflicting following the Archimedean interpretation of numbers as distances, https://en.wikipedia.org/wiki/Archimedean_property.

1) We may follow Archimedes in interpreting numbers as distances. Right?
2) Than we may interpret the equation 5 + 5 = 10 as: starting from a given point and traveling in two steps of 5 units we’ll cover a distance of 10 units. Right?
3) We may write 0.999... as: 0.(9). Right?
4) The notation 0.(9) represents the infinite sequence: 0.9, 0.99, 0.999, etc. Right?
5) The difference between 1 and 0.99 and 0.9 = 0.09, between 0.999 and 0.99 = 0.009, between 0.9999 and 0.999 = 0.0009 etc. Right?
6) These differences can be written as the decreasing sequence 0.09, 0.009, 0.0009, etc, or: as 9.10E-2, 9.10E-3, 9.10E-4, etc. (E means Exponent) Right?
7) If reals are interpreted as distances and 0.(9) as representing the sequence: 0.9, 0.99, 0.999, etc, the notation 0.(9) = 1 must mean: starting from 0.9 we can cover the distance 0.9 to 1 in ever decreasing steps: 9.10E-2, 9.10E-3, 9.10E-4, etc. Right?
8) Every possible negative power of 10 must be a negative natural number. Right?
9) Every real 10E-n must be a real > 0. (If 10E-n = 0 should be true, 1 and thus 10En too, had to be 0.) Right?
10) Because every real 9.10E-n > 0, it is impossible to merge with 1 from 0.9 in decreasing steps: 0.09, 0.009, 0.0009, etc. Right?
11) But if we cannot reach 1 by traveling from 0.9, to 0.99, to 0.999, etc, the expression 0.(9) = 1 cannot be true. Right?
N.B.1: I suggest to rewrite the notation 0.999... = 1, in agreement with the Archimedian interpretation of numbers as distances as: 0.999... ≠ 1 nearing but not reaching a limit 1.
N.B.2: Zeno of Elea showed 2300 years ago, that 1 could not be reached in decreasing units 0.5, 0.25, 0.125, etc, see: “The Dichotomy” in: https://plato.stanford.edu/entries/paradox-zeno/.
Can there be an infinite number and an actual infinity?
1) The notation 0.(9) means an infinite multitude of nines to the right of the decimal point. Right?

But can there be a sequence of nines that is ‘actual infinite’, that is: a sequence in which the infinite multitude of nines is really listed, no number nine exempted? Let us see, with the infinity of the naturals as an example.

2) Euclid’s Theorem, https://en.wikipedia.org/wiki/Euclid%27s_theorem, proofs by a a reductio ad absurdum , https://en.wikipedia.org/wiki/Reductio_ad_absurdum, that the assumption of a largest prime is absurd; illogical. Right?
3) As primes are uneven natural numbers and any even number is uneven number + 1, starting from this Theorem we may say that the assumption of a largest natural must be illogical too. Right?
4) But if there cannot be a largest natural in the infinite sequence: 1, 2, 3, etc, this sequence cannot have an upper boundary. Right?
5) In that case this infinite sequence of naturals must be understood as an ever in number growing sequence of elements without an upper boundary. Right?
6) Than ‘infinite’ in the infinite sequence of naturals can exclusively and only be understood as:‘ever growing without an upper boundary’. Right?
7) The expression: ‘infinite sequence of elements’ must than be interpreted as an: ‘ever growing sequence of elements without an upper boundary’. Right?
8) In an ‘ever growing sequence without boundary’ we cannot make the, without boundary ever growing multitude of elements, invariable by assigning a number to it. Right?
9) So an infinite: without upper boundary ever growing multitude of elements, can never loose its property: ever growing in elements without an upper boundary, by assigning a natural or hypothetical infinite number to it. Right?
10) That's why we may assign to this ever growing multitude of elements, the notion ‘infinite number’, something like: Aleph, Omega, or ∞. Right?
11) But assigning an infinite number to such a without upper bound ever growing multitude of elements like nines in: 0.(9): 0.9999999999,etc, or naturals in: 1, 2, 3, etc, cannot have any effect on the properties of this multitude. Right?
12) As this sequence is ever growing without upper boundary in elements it must than be true that, during the time you need to say: “An infinite number is represented by the symbol∞”, it could be that the number of nines on the moment you have said it, has become: ∞ + 2.10E9 and 5 minutes later ∞ + 302.10E9. Right?
13) ((Well, that’s why I think that the notation (0.9)9 does not equal nonsense. Right? (But I'll never use this notation 0.(9)9 again.))
14) That's why starting from Euclides Theorem and its consequences there cannot be an ‘actual infinite’, that is: a sequence in which the infinite multitude of nines is really listed, no number nine exempted. Right?
15) The assumption of a number 'infinite' would only be logically defensible if an 'actual' infinite could exist. Right?
16) So because, seen in the light of Euclid's Theorem and its consequences the assumption of an actual infinite is illogical, the assumption of a infinite numder must be illogical too. Right?

And for the record: and than a mathematical theory which is based on an axiom like: there is a number ’infinite’, or: actual infinity is possible, must be illogical, because such a theory is from its beginning in conflict with the consequences of Euclid’s Theorem. If I'm right in this, this position must have his impact on mathematics. For example: Cantor’s Diagonal Argument cannot be true than, because of the interpretation of ‘infinite’ as: ‘ever growing in elements without upper boundary (or shorter: without a limit) in elements’ This Argument can than be paraphrased as a a composition of the infinite, ergo ever growing, sequence of naturals: 1, 2, 3, , etc, and the infinite: ever growing sequence of reals < 1 in random order: 0.6, 0.33, 0.14, etc. From this sequences the one-to-one correspondence: [1, 0.6], [2, 0.33], [3, 0.14], etc, can be prepared. Ergo: both sequences must have the same cardinality.Antonboat (talk) 20:43, 18 December 2017 (UTC)

May I point you to your habits of obstinately denying
- to keep up the difference between a whole "sequence" (established in (4)) and "single items" of this sequence, as used in other parts of your arguments
- to distinguish between the "infinity" as defined/used in the context of of this article (limits!), and the philosophical "infinite multitude" you refer to. BTW and for the records, there is no axiom that "infinity is a number" within real analysis.
- to accept that you are free to define "your infinity" to your likings, without this having any influence on the truths in other axiomatic systems. (The article to which this talk page belongs is within the realm of the standard real numbers.)
and -most of all-
- that each real number, as addressed in this article, is a full equivalence class of all sequences "converging" (involves one well defined conceptualization of "infinity"!) to the same "limit". The sequence 0.(9), according to your definition, denotes just one of these sequences, and its limit 1 is, sensibly, the representative for this very equivalence class, containing "infinitely" many other sequences having the same limit, and all of these are denoted by the same representative. (This is similar to all fractions (ax/bx) being contained within one equivalence class, which itself is the one rational number, all these fractions are equal to.)
I consider it as fully de rigeur that you deny for philosophical reasons the existence of an "infinity" as referred to within real analysis, but then you must also oppose to large parts of modern math, which the vast majority of contemporary mathematicians consider sound, useful, and interesting, even when contradicting some puristic speculations.
It is to me not useful to deny the existence of alien axiomatic systems, but within real analysis your argumentation does not hold, and the thread title is a wrongness in the first claim. Purgy (talk) 10:08, 19 December 2017 (UTC)
Dear Purgy. Thank you for your comments. You write: “May I point you to your habits of obstinately denying. “ Please do, for that’s what an “argument” page is meant for. And it could be posssible that I have “habits of obstinately denying¨, indeed. But can you speak of ‘denying’ if you come to a certain conclusion by means of : ‘logical arguing’?'
Sub a) you write you think its obstinately denying that I: “keep up the difference between a whole "sequence" (established in (4)) and "single items" of this sequence, as used in other parts of your arguments”. Why should I keep up this difference? 4) reads: 4) The notation 0.(9) represents the infinite sequence: 0.9, 0.99, 0.999, etc. What’s wrong than by stating: 5) The difference between 1 and 0.99 and 0.9 = 0.09, between 0.999 and 0.99 = 0.009, between 0.9999 and 0.999 = 0.0009 etc. Do you think that statement is not true? Please proof this. That’s what my argumentation in steps 1), 2) and more, is meant for. I tried, by what I thought to be: my logical arguing, to come to the conclusion that the expression 0.(9) = 1 and thus that 0.9 + 0.09 +0.009, etc, = 1 could not be true. Please show me where I went wrong in my arguing that 0.9 + 0.09 +0.009, etc, must be < 1.
Sub b): “to distinguish between the "infinity" as defined/used in the context of of this article (limits!), and the philosophical "infinite multitude" you refer to.” I came, by what I thought to be : my logical arguing, in the second part: Can there be an infinite number and an actual infinity? In step 1) to 6) to the conclusion that ‘infinite’, seen in the light of Euclid’s Theorem and what I thought its consequences for the properties of the naturals, to conclusion 6) : “Than ‘infinite’ in the infinite sequence of naturals can exclusively and only be understood as:‘ever growing without an upper boundary’. Right?” From Euclid’s Theorem is said on Wikipedia: : Euclid's theorem is a fundamental statement in number theory that asserts that there are infinitely many prime numbers. There are several well-known proofs of the theorem.” So I had a sound mathematical foundation to start from, I thought.
”BTW and for the records, there is no axiom that "infinity is a number" within real analysis.” (Do you know that in Dutch by writing “BTW” we mean: Belasting Toegevoegde Waarde?) In the second part of my note, after 16), I write “And for the record: and than a mathematical theory which is based on an axiom like: there is a number ’infinite’, or: actual infinity is possible, must be illogical, ...” Here is spoken of ‘a’ mathematical theory.
Sub c) “- to accept that you are free to define "your infinity" to your likings, without this having any influence on the truths in other axiomatic systems. (The article to which this talk page belongs is within the realm of the standard real numbers.)” No Purgy, I didn’t feel free. I chose to start from a sound mathematical foundation: Euclid's Theorem, see sub b) and found a different interpretation of ‘infinite’ as property of the repeating nine: 0.(9).
Sub d) “and -most of all-
- that each real number, as addressed in this article, is a full equivalence class of all sequences "converging" (involves one well defined conceptualization of "infinity"!) to the same "limit". The sequence 0.(9), according to your definition, denotes just one of these sequences, and its limit 1 is, sensibly, the representative for this very equivalence class, containing "infinitely" many other sequences having the same limit, and all of these are denoted by the same representative. (This is similar to all fractions (ax/bx) being contained within one equivalence class, which itself is the one rational number, all these fractions are equal to.)” Which definition of ‘infinite’ do you use here? If Euclid’s Theorom is right and the argumentation of the consequences of it is solid, there can be only one interpretation: an infinite repeating of the nine without upper boundary (I think).
Sub e) “I consider it as fully de rigeur that you deny for philosophical reasons the existence of an "infinity" as referred to within real analysis, but then you must also oppose to large parts of modern math, which the vast majority of contemporary mathematicians consider sound, useful, and interesting, even when contradicting some puristic speculations.”
According to: https://www.merriam-webster.com/dictionary/de%20rigueur, ‘de rigueur’ means: prescribed or required by fashion, etiquette, or custom. No Purgy, I’m a man of 75, who strives to be polite, friendly and reasonable and critical. But I’m too old to follow whatever fashion. I do not deny for fashionable reasons the ‘infinite’ as you see it. I see, following Euclid an ‘infite’ that’s neither your vision on it, nor that of the vast majority of contemporary mathematicians.” As far as I know mathematics is a discipline that is founded on well-established starting points, and I see Euclid’s theory as one of them. Please show me that ’infinite’ the way I see is a puristic speculation.
Sub f) “It is to me not useful to deny the existence of alien axiomatic systems, but within real analysis your argumentation does not hold, and the thread title is a wrongness in the first claim.” Please show me, by logically reasoning the number of the argument where, according to you, I was wrong. — Preceding unsigned comment added by Antonboat (talkcontribs) 12:34, 20 December 2017 (UTC)
@Antonboat: Do you know the definition of a "sequence"? I have debunked the rest of what you said below.--Jasper Deng (talk) 12:45, 20 December 2017 (UTC)
According to Wikipedia "A sequence can be thought of as a list of elements with a particular order". Why did you ask this? Regarding your comment:"I have debunked the rest of what you said below." What rest have I said below?Antonboat (talk) 13:07, 20 December 2017 (UTC)
@Antonboat: Hence why I think you lack the necessary prerequisites for having a mathematical discussion. You need to know the rigorous definition of a sequence as a function from the natural numbers to some other (nonempty) set. A net is a generalization to where the domain need not be the natural numbers. And as for my "debunked" phrase, I meant that I debunked your initial comment in this section with the comment I made below.--Jasper Deng (talk) 13:10, 20 December 2017 (UTC)
@Antonboat: Sorry, TL; DR: what you wrote above is not even wrong, for the reasons stated by Purgy.--Jasper Deng (talk) 11:36, 20 December 2017 (UTC)
And if I may add to what Purgy said:
  1. Before you can have a conversation about the real numbers at this level of abstraction, you have to work with a construction of the real numbers, many of which use an equivalence relation of some sort.
  2. Point 7 in the first list of arguments is a bona fide example of hand-waving. You need a meaningful and rigorous definition of "covering a distance" before you can attempt to make an argument with it.
  3. Point 8 above is blatantly incorrect. Natural numbers can only be integers and you are at the minimum talking about rational numbers.
  4. Point 10 is also hand-waving, because you did not define "merge with".
  5. Hence point 11 is meaningless.
  6. In your second list, points 2 and 3 do not need Euclid's theorem to prove. The natural numbers being an infinite set follows from the very definition of finiteness: a set is finite if and only if it can be put in bijection with a bounded subset of the natural numbers.
  7. Point 6 in your second list is hand-waving. A rigorous definition of "infinite" when referring to the cardinality of a set is that there exists no surjective function from a bounded subset of the natural numbers to that set.
  8. In light of the previous point of mine, your point 10 is a misinterpretation of the accepted meaning of ordinals like aleph-naught.
  9. For point 11, as has been emphasized repeatedly (even by yourself), infinity is not a natural number. It is incorrect to assert that adding an "infinity" to the natural numbers will preserve the properties of the natural numbers defined as-is - notably, as you yourself have pointed out, such a new number system could not enjoy the Archimedean property that you emphasize so much.
  10. For point 12, you are once again hand-waving. The notion of any number plus infinity is not meaningful unless you choose a definition of how you are extending the addition operation to operate on "infinity".
  11. For your remaining points, whether "infinity" is considered to be a number per se is a matter of convention. However, it is a well-defined concept.
I don't mean to come off as rude, but in order to talk with mathematicians, you need to, in a sense, speak the language of mathematicians. Learn to make everything you say well defined. This is all in addition to what Purgy said above, and not in conflict with or in place of it.--Jasper Deng (talk) 12:10, 20 December 2017 (UTC)
Dear Jasper deng. I have no time now to respond to your comments. For now, you write:"I don't mean to come off as rude, but in order to talk with mathematicians, you need to, in a sense, speak the language of mathematicians." You may be rude to me I'm a psychologist and used to that, I earned my money with it. But what's your intention with your rudeness? Making me angry, afraid, unsure? Are you unable to respond politely? Than I have to accept that. But why not try to show me with logical arguments instead of rude words that I am wrong. Please try it.Antonboat (talk) 13:34, 20 December 2017 (UTC)
@Antonboat: I just gave you the exact logical flaws in your argument. And I would love to "fix" your argument but at this time, it is insufficiently rigorous for me to follow your path. The reason for the dissection is to hopefully give you a sense of the standards expected for a discussion of this sort, because we always love seeing new ideas - but only when those ideas are stated in an adequate framework, which yours unfortunately is not. Part of it, I guess, is that I'm irked when I see a highly flawed argument passed off as "logical reasoning". If you are unsure of the footing of your argument, then by all means ask the necessary questions and do the necessary research. Wrongly claiming that I'm "very wrong" is going to be met by showing the exact opposite; pot meet kettle.--Jasper Deng (talk) 13:38, 20 December 2017 (UTC)
Dear Jasper deng. These are my comments on the comments you made in Jasper Deng (talk) 12:10, 20 December 2017 (UTC). Thank you for your evaluation that my note you comment on is ‘not even wrong’.
Sub 1: You write: “ Before you can have a conversation about the real numbers at this level of abstraction, you have to work with a construction of the real numbers, many of which use an equivalence relation of some sort.” Why must I do that in my note while it isn’ t done by the author of Wikipedia “0.999...”., from which article I copied the row/series/sequence 0.9, 0.99, 0.999, etc, as a representation of 0.(9), and by the wikipedians which reacted on my notes? For example Double sharp wrote to me: “Sequence 0.999... can be defined as the limit of that sequence (0.9, 0.99, 0.999, ...), i.e. a number separate from the sequence which these terms get as close as you want to (but might not reach). Double sharp (talk) 12:52, 3 December 2017 (UTC). And above all, why do I have to do that when the numbers I used in the first part of my note are repesented by distances? Because I’m no mathematician? Or are you nitpicking here?
Sub 2: You write: “Point 7 in the first list of arguments is a bona fide example of hand-waving. You need a meaningful and rigorous definition of "covering a distance" before you can attempt to make an argument with it.” Is “travel” better instead of “cover”? But Is such a meaningful and rigorous definition of "covering a distance" needed here. Would this lead to different result than I found in 7)?
Sub 3: You write: “Point 8 above is blatantly incorrect. Natural numbers can only be integers and you are at the minimum talking about rational numbers.” Please read this line once again but now good! I’m speaking there of the negative powers of 10 in line 7) and these are all negative natural numbers.
Sub 4: You write: “Point 10 is also hand-waving, because you did not define "merge with".“ You’re right, I could not find the right word for the Dutch expression: ‘in elkaar opgaan’. Is ‘unite’ a better word for a situation in which the distance between a real seen as a distance and 1 = 0? And, is the conclusion in 10) wrong?
Sub 5: You write: “Hence point 11 is meaningless.” But if I write instead of: “But if we cannot reach 1 by traveling from 0.9, to 0.99, to 0.999, etc, the expression 0.(9) = 1 cannot be true.”: “But as there cannot be a real in the row 0.9, to 0.99, to 0.999, etc, that has a distance = 0 to 1 we can never unite with 1 traveling from 0.9, to 0.99, to 0.999, etc.”
Sub 6: You write: “In your second list, points 2 and 3 do not need Euclid's theorem to prove. The natural numbers being an infinite set follows from the very definition of finiteness: a set is finite if and only if it can be put in bijection with a bounded subset of the natural numbers.“ So 2) and 3) are true?
Sub 7: You write: “Point 6 in your second list is hand-waving. A rigorous definition of "infinite" when referring to the cardinality of a set is that there exists no surjective function from a bounded subset of the natural numbers to that set.” And than this unbounded set of natural numbers must be seen as: ever growing without an upper bound, isn’t it?
Sub 8: You write: “In light of the previous point of mine, your point 10 is a misinterpretation of the accepted meaning of ordinals like aleph-naught.” Oh no, I just named it and didn’t even try to define it. But if I should try it I would start this, by the definition of an ordinal which is given in: https://en.wikipedia.org/wiki/Ordinal_number in which is written: “Perhaps a clearer intuition of ordinals can be formed by examining a first few of them: as mentioned above, they start with the natural numbers, 0, 1, 2, 3, 4, 5, … After all natural numbers comes the first infinite ordinal, ω, and after that come ω+1, ω+2, ω+3, and so on.” And it follows from that definition that all the ordinals assigned to an infinite multitude of naturals wihout an upper boundary must also be naturals.”
Sub 9: You write: “For point 11, as has been emphasized repeatedly (even by yourself), infinity is not a natural number. It is incorrect to assert that adding an "infinity" to the natural numbers will preserve the properties of the natural numbers defined as-is - notably, as you yourself have pointed out, such a new number system could not enjoy the Archimedean property that you emphasize so much.“ I did not argue that there. I’m saying there: if you wish, you may assign an infinite number tot this ever without upper bound growing number. But in such way you can not change the property of such an ever growing multitude of nines or naturals. N.B.: I never emphasize in this note the Archimedean property. Following Archimedes I interpret numbers as distances, see 1) in the first part of my notation.
Sub 10: You write: “For point 12, you are once again hand-waving. The notion of any number plus infinity is not meaningful unless you choose a definition of how you are extending the addition operation to operate on "infinity". Oh no my dear Jasper deng. Read 12) once again but now good! What I’m saying there is: you may assign any number or symbol to an infinite without upper boundary growing multitude, but that must be meaningless, because after that, this multitude is stil growing, because of Euclid’s Theorem and its consequences, as seen above, see second part of this note 2) to 7).
Sub 11: You write: “For your remaining points, whether "infinity" is considered to be a number per se is a matter of convention. However, it is a well-defined concept. “ Why don’t you comment on 13) to 16) . Do you agree with them? If not, I invite you here to comment on this items.
Dear Jasper deng. What I meant with this note is a concept to create a logical framework to proof the idea a) that 0.999... = 1 could not be right if reals would be seen, following Archimedes as distances, and b) that starting from Euclides Theorem and his consequences there could not be an actual infinite and no infinite numbers.
Evaluation of your comments. You have shown me that when I used some expressions, they were not always in the right mathematical terms. Thank you for that, but I suspected that already, see above. You write: “ I don't mean to come off as rude, but in order to talk with mathematicians, you need to, in a sense, speak the language of mathematicians. Learn to make everything you say well defined. “ What you say here is, I think, not fair arguing. What would you say if I told you that your arguments have no value for me because they were not written in perfect Dutch? Would I be rude to you if I should say that you’re often “nitpicking” in your comments by: fussy or pedantic fault-finding? In my opinion you do this in a effort to slalom the really important ideas in this note. In my opinion you barked loudly, but you didn’t bite me. Please bite me.
- Show me that the infinite multitude of naturals has some upper boundary and may be seen as a set which is actual infinite.
- Bite me and show me that aleph naught can not be a natural number.
- Bite me and show me that we may not interpret the infinite of the natural numbers as: an ever growing multitude without an upper boundary.
- Please bite me deep, and show me that Cantor’s Diagonal Argument can be true, if ‘infinite’ is interpreted as: ‘ever growing in elements without upper boundary (or shorter: without a limit) in elements’ .
-Bite me still deeper by showing me that my parafrase of this Argument that it must be seen as a a composition of the infinite, ergo ever growing, sequence of naturals: 1, 2, 3, , etc, and the infinite: ever growing sequence of reals < 1 in random order: 0.6, 0.33, 0.14, etc., and its conclusions: “From this sequences the one-to-one correspondence: [1, 0.6], [2, 0.33], [3, 0.14], etc, can be prepared. Ergo: both sequences must have the same cardinality”, must be wrong. From this sequences the one-to-one correspondence: [1, 0.6], [2, 0.33], [3, 0.14], etc, can be prepared. Ergo: both sequences must have the same cardinality.", is not true.
Shakespeare wrote: “What's in a name? That which we call a rose by any other name would smell as sweet.” , https://www.brainyquote.com/quotes/william_shakespeare_125207. Does the sequence 1, 2, 3, etc change in properties because I call it a series or a row? Don’t you understand what I mean if I write: “7) If reals are interpreted as distances and 0.(9) as representing the sequence: 0.9, 0.99, 0.999, etc, the notation 0.(9) = 1 must mean: starting from 0.9 we can cover the distance 0.9 to 1 in ever decreasing steps: 9.10E-2, 9.10E-3, 9.10E-4, etc.” If you think the word “cover” is not the right word here, why don’t you give me a better, without changing the meaning of this sentence.Antonboat (talk) 09:39, 21 December 2017 (UTC)
@Antonboat: In one word: rigor. If you're going to resist and deny the need to be rigorous, you'll get nowhere, and we will not entertain having this conversation. The simple fact is that what you are saying is too vague for me to infer anything meaningful. Wikipedia's articles are not meant to always give a fully rigorous definition in the first sentence, but (ideally) such a definition is always found later in the article, and because you are trying to make an argument here, you are required to use mathematical language. Because the simple fact is that I could infer nothing meaningful from what you are saying. Although the notion of "distance" is formalized by the notion of a metric space, in the physical world one probably cannot meaningfully talk about "distance" shorter than the Planck length, and at this time, the way you use the word "distance" is too hand-wavy for me to infer anything meaningful from it. Whether real numbers should be thought of as representing "distance" is a matter of convention, but what is not acceptable is using it in such a vague and imprecise manner. This is mathematics, not Dutch, so the nitpicking here is going to be mathematical in nature. And the nitpicking is necessary because right now, you are saying absolutely nothing meaningful. This is completely fair because the burden is on you, as the arguer, not me, to make a coherent argument.
I also think you completely miss the notion of convergence of a sequence, along with its relation to "infinity" and what Purgy has said below about a sequence and its limit, and that before continuing further you must familiarize yourself with the precise definition of convergence. What Double sharp said there was meant to provide you with intuition, but as a definition, is hand-waving and insufficient for an argument of this sort. While it is true that no term of the sequence is exactly 1, its limit is 1 and we define the notation 0.(9) to denote the limit of the sequence, not any particular term. That argument has been presented numerous times on this very talk page and soundly debunked every single time: you are conflating members of the sequence with the sequence's limit. And no, Cantor's diagonal argument completely refutes your above argument attempting to show that the reals are countable: no matter how you try to enumerate the reals I can give you another that wasn't in your sequence. I also think you miss the point that there exists no sequence that enumerates all the real numbers - did you actually read the article on the argument? Both the reals and natural numbers are sets of infinite cardinality, but the former is far bigger than the latter, and you are making a circular argument by assuming a priori that the reals can be represented as a sequence in the first place. It is also not up to me to guess what you are trying to say by getting "closer" - your sentence had no meaning in the first place, so there is nothing I can do to "correct" it.
I accept wholeheartedly that (as precisely stated), the cardinality of the set of natural numbers is infinite, namely aleph-naught. But you cannot operate on infinity like a number. Thus your "∞ + 2.10E9" is a meaningless expression. "Ever growing without upper bound" is hand-wavy and you should wean yourself off using that phrase, since while it is acceptable as a way of gaining intuition, it is not acceptable as a definition of "infinite" - I already gave you a rigorous one and you should try to embrace it. The existence of the concept of "infinity" does not, as explained to you repeatedly, contradict Euclid's theorem. That theorem says nothing about extending the naturals to include infinite members. Your point 9 above is in direct contradiction to what you have said here. And ultimately, if 0.(9) isn't 1, then what is it?
I could go on and on about all the problems with your argument, but I will stop here. It's hard to tell if you have been trying to embrace the requirements of talking about this subject. You have been told numerous times that absent sufficient rigor, we are left scratching our heads about what you are trying to say. I recall you, as a psychologist, wanted to see our thought process here. We could give you an informal description of it, but I sense you want the exact steps. However, for the latter, we are going to talk in the language we speak, and it's up to you, not us, to ensure you understand it.--Jasper Deng (talk) 17:46, 21 December 2017 (UTC)
Dear Jasper deng. Thank you for your comments.
You write: “While it is true that no term of the sequence is exactly 1, its limit is 1 and we define the notation 0.(9) to denote the limit of the sequence, not any particular term.” That’s what I meant with my suggestion: N.B.1: I suggest to rewrite the notation 0.999... = 1, in agreement with the Archimedian interpretation of numbers as distances as: 0.999... ≠ 1 nearing but not reaching a limit 1. “ That would , in my opinion, prevent a lot of comments, to the Argument page in the future. (I’ve deleted the words : “in agreement with the Archimedian interpretation of numbers as distances“ here because they are not relevant for the definition of 0.999....)
You write: “That argument has been presented numerous times on this very talk page and soundly debunked every single time: you are conflating members of the sequence with the sequence's limit. “ No I did not and I do not, see my definition above..
You write: “And no, Cantor's diagonal argument completely refutes your above argument attempting to show that the reals are countable: no matter how you try to enumerate the reals I can give you another that wasn't in your sequence.” Yes and I can give you a natural number to serve as an ordinal for every real that you create in Cantor's diagonal argument, starting from Euclid’s Theorem and its consequences.
You write: “I also think you miss the point that there exists no sequence that enumerates all the real numbers - did you actually read the article on the argument? “ I did read a lot of articles even in German and even the interpretation of Douglas Hofstadter in: “Gödel Escher Bach”. I never could have missed this point because I see the infinite series of reals as ever growing without an upper boundary. Following Euclid’s Theorem and its consequences, I argued in this note: “Cantor’s Diagonal Argument cannot be true than, because of the interpretation of ‘infinite’ as: ‘ever growing in elements without upper boundary (or shorter: without a limit) in elements’ This Argument can than be paraphrased as a a composition of the infinite, ergo ever growing, sequence, read here: “series” of naturals: 1, 2, 3, , etc, and the infinite: ever growing sequence, read here instead: “series” of reals < 1 in random order: 0.6, 0.33, 0.14, etc. From this sequences /series the one-to-one correspondence: [1, 0.6], [2, 0.33], [3, 0.14], etc, can be prepared. Ergo: both sequences/series must have the same cardinality.”
You write: “Both the reals and natural numbers are sets of infinite cardinality, but the former is far bigger than the latter, and you are making a circular argument by assuming a priori that the reals can be represented as a sequence in the first place.” I've never said that the reals could be represented by a sequence/series of all!! the reals. From Euclid’s Theorem follows that there can’t be no largest natural number, thus although the number of natural numbers and of the reals are infinite: ever in mutitude growing without an upper boundary, the logical consequence of it must be that there cannot be an infinite series of all the naturals none exempted, and nor of the reals. You write: “you are making a circular argument by assuming a priori that the reals can be represented as a sequence in the first place.” No, I didn’t , but Cantor, which I hold in high esteem because of its pioneering research concerning infinity, presents (not: 'represents’ Jasper!) in his Argument the reals as a series in random order.
As an aside: You write: “ ... in the physical world one probably cannot meaningfully talk about "distance" shorter than the Planck length, ... ” In: https://en.wikipedia.org/wiki/Planck_length, is said “Much like the rest of the Planck units, there is currently no proven physical significance of the Planck length.” If this unit is real and > 0 it can be divided. If it’s pure hypothetical it can be illusive, and such units you can only divide by illusive dividing.
An extremely short summary of all my contributions on this Argument page would be: Antonboat is arguing that there are as many possible nines and thus naturals, as there are reals. I therefore suggest that our communication will be limited to the question of whether there are just as many nines, so naturals, as reals.
I must say I appreciate your way of communicating very much, but not to much hand-waving please, it might hurt your wrist.
Merry Christmas to you to and to all the Wikipedians that commented on my contributions.Antonboat (talk) 15:51, 22 December 2017 (UTC)
@Antonboat: You are free to disagree with the definition of 0.999... as the limit of that sequence, but then in order to even compare it to a given number you have to still assign a meaningful value to it. Hence is a meaningless statement under what you are trying to suggest. Regarding distances, the Pauli exclusion principle still sets limits on how much distance we can measure. And you are yet again flatly incorrect about Cantor's diagonal argument. I repeat, the reals cannot be enumerated by a single sequence, and assuming otherwise is plainly circular reasoning - you are basically assuming that all infinite cardinalities are equal, which is patently false. In mathematics, you have to clearly, succinctly, and precisely state every assumption you make, no exceptions. In fact you can't enumerate the reals with countably many sequences either. You cannot claim that you can provide a natural number for every real number; in the particular case of the number Cantor constructs as a counterexample, the presumption is that you have already mapped every natural number to a real number, so you are no longer free to then assign another natural number to that counterexample. If you do assign a particular natural number then you have to give up the assignment you had earlier made for it; either way you failed to cover all the real numbers. Also, strictly speaking, we are not talking about infinite series here with your particular example here, we are talking about infinite sequences. If you want to use the reals as the indexing set rather than the naturals, then we can't talk about sequences anymore; we have to talk about nets, and I suggest you avoid that given your current state of confusion.--Jasper Deng (talk) 20:04, 22 December 2017 (UTC)
Dear Jasper Deng. You write: “You cannot claim that you can provide a natural number for every real number; in the particular case of the number Cantor constructs as a counterexample, the presumption is that you have already mapped every natural number to a real number, so you are no longer free to then assign another natural number to that counterexample.”
But yes I claim I can. Give me any real 10E-n as a starting point and I’ll show you how you can do that.
Indeed Cantor presumed, but did not prove!, in his diagonal argument, that he had already mapped every natural number to a real number. Since when is an assumption! of something in science accepted as a “proof”?Antonboat (talk) 12:08, 2 January 2018 (UTC)
No you can't. And Cantor was probably not talking about what you think he was, and if he was, then he was talking about the nonexistence of such. When I say "presumption", I am talking about the hypotheses Cantor starts with before beginning his proof by contradiction (which always starts with an assumption, namely the negation of what you are trying to prove).--Jasper Deng (talk) 20:08, 2 January 2018 (UTC)
The gold standard for a proof nowadays is that not only has it been peer reviewed but that one has passed it though Automated proof checking. This does not stand for hand waving type arguments or fuzzy thinking. This starts off with a set of axioms and rules for deduction and gets rigorous proofs. Different axioms and rules can led to different things but there is a pretty much agreed part for everyday work rather than testing what can happen. The standard real number system where 0.999... = 1 and Ordinal numbers which include various infinities are part of that, and it is extremely unlikely that someone is going to come along and find some contradiction in them. So what I see you as basically saying is you don't like people bandying around infinity as a real thing. Well that's tough. Mathematicians find the concept works fine and is useful. As David Hilbert said "No one shall drive us from the paradise which Cantor has created for us." Dmcq (talk) 17:05, 20 December 2017 (UTC)
Dear Dmcq. You write: “The standard real number system where 0.999... = 1 and Ordinal numbers which include various infinities are part of that, and it is extremely unlikely that someone is going to come along and find some contradiction in them.” I’m sure that what you say here is true. But the theories that contain different infinities must be based on the axiom: there is an actual infinity. Seen in the light of an mathematically accepted proof, like Euclid's Theorem and the consequences i've been described, see for example in: Can there be an infinite number and an actual infinity?, the points 1) to 7) the natural numbers must be an ever, without an upper boundary, growing number. According to mathematicians here in Holland, mathematics cannot manage such a concept, but very well the axiom: There is an actual infinity. But this axiom is and stays in conflict with the consequences of Euclid's Theorem, whether you agree with it or not.Antonboat (talk) 09:16, 22 December 2017 (UTC)
@Antonboat: Nope. As Purgy has said before, "infinity exists" is not an axiom that defines the real numbers and without a definition of "infinity", is not a well-defined statement. The closest precise concept is the axiom of infinity (yes, please read that article in full: it doesn't say what you are trying to say). And yet again, you are plainly incorrect in whatever you might mean by "this axiom is and stays in conflict with the consequences of Euclid's Theorem". The theorem, after all, already uses the notion of an infinite set, and in particular is a statement that the prime numbers form such a set. Your statement "the natural numbers must be an ever, without an upper boundary, growing number." is absolutely meaningless, and appears to conflate the set of natural numbers with the members of that set. This is why we can't infer anything meaningful from your above comment(s).--Jasper Deng (talk) 09:22, 22 December 2017 (UTC)
No jasper deng, if you accept Euclid’s Theorem as proof that the assumption of a largest prime is absurd: illogical, you have to accept the same for the naturals. In that case a set of all!! naturals is impossible. Again: Merry Christmas..Antonboat (talk) 16:44, 22 December 2017 (UTC)
@Antonboat: You are for the umpteenth time flatly wrong about this, unless you can prove that the axiom of infinity is inconsistent with the other axioms of ZFC set theory.--Jasper Deng (talk) 20:04, 22 December 2017 (UTC)
Dear Jasper Deng You write: “You are for the umpteenth time flatly wrong about this, unless you can prove that the axiom of infinity is inconsistent with the other axioms of ZFC set theory.” I’ m sure that this axiom is not in conflict with the other axioms. But can you prove me that the axiom of infinity is true?Antonboat (talk) 12:22, 2 January 2018 (UTC)
An axiom cannot be proven or disproven. But you can't have it both ways. The natural numbers are by definition the object (a set in particular) whose existence is asserted by the axiom of infinity.--Jasper Deng (talk) 20:00, 2 January 2018 (UTC)
As a newcomer to the discussion, I think I may help you with your intuitions on infinity, as I struggled with the concept for some time as well. A similar case for me was the concept of the imaginary number i, which is defined as the square root of -1, a negative number. While it's true that it's impossible to calculate the imaginary unit from the basic operations introduced before for real numbers (addition and multiplication), that doesn't prevent mathematicians from treating it as a mathematical object and study their properties, which happen to be quite useful in practice. As long as no contradictions appear by declaring that such object exist, you're creating a new area of mathematics that is not incompatible with the previous principles, but that is an extension of them.
Another example is with the assertion that "the sum of the angles of a triangle is 180º"; if you study what happens if you declare that this assertion is false, you get the concept of non-euclidean geometry. Infinity should be handled using the same intuitions: its existence does not follow from the basic operations of finite and finitely enumerable numbers, but these operations don't prevent its existence either; therefore, the mathematics that appear from declaring that such object exists are useful and worthy of study.
Under this light, your reasoning above ("if there's no largest natural, then a set containing all naturals is impossible") is a non-sequitur: the consequent doesn't follow from the premise; the only consequence from that premise is that such set is not finite, not that it doesn't exist.
In fact you can define such a set in a rigurous way. If I remember my basics from early programming theory courses, a set can be defined by intension by defining a test function - a property that all its members possess, and only its members. In the case of naturals, that property is "it can be calculated by repeatedly applying the operation "successor" to 0 a finite number of times". No mater how high is the natural number you give me, I theoretically could confirm that it belongs to the naturals by applying my function. Then I can define the set of all Naturals as "the set of all objects that satisfy my above test function (and only those)". In this case, the equivalent of Euclid's Theorem for natural numbers shows is that this set can not possibly be finite (if you say that any particular natural number n is the last one in the set, I can create a new natural number larger than that: n' = successor(n), which is a natural because it passess the test function). Diego (talk) 20:16, 22 December 2017 (UTC)
Dear Diego.You write: “Under this light, your reasoning above ("if there's no largest natural, then a set containing all naturals is impossible") is a non-sequitur: the consequent doesn't follow from the premise; the only consequence from that premise is that such set is not finite, not that it doesn't exist.” In my opinion the consequence of Euclid’s Theorem is that as there can’t be a largest natural, the infinite set of naturals must be interpreted as a finite set of naturals to which boundlessly are added new, unique naturals.Antonboat (talk) 11:51, 2 January 2018 (UTC)
You said it: that's an opinion, not a logical inference. Thus, the consequence does not result from a formal proof from the steps before it. In mathematics, every step in a proof must follow from the axioms, forming an unbroken chain of simple and solid reasoning steps, using a well-defined set of inference rules; there's no place to opinion (other than the act of choosing the initial set of axioms). Diego (talk) 22:55, 2 January 2018 (UTC)
@Antonboat: I should also add that I mean it when I ask you to drop your hand-wavy "definition" of "infinity" as "ever growing without an upper boundary". The set of real numbers between 0 and 1 exclusive is a bounded subset of the real numbers and yet is of infinite cardinality. It seems to be blinding you to what mathematicians mean by "infinite": a given set is said to be of infinite cardinality precisely if there exists no surjective function from any bounded subset of the natural numbers to that set.--Jasper Deng (talk) 09:49, 23 December 2017 (UTC)
Dear Jasper Deng. You write: “The set of real numbers between 0 and 1 exclusive is a bounded subset of the real numbers and yet is of infinite cardinality. It seems to be blinding you to what mathematicians mean by "infinite": a given set is said to be of infinite cardinality precisely if there exists no surjective function from any bounded subset of the natural numbers to that set.” A bounded subset of the real numbers that could be of infinite cardinality, isn’t that a contradictio in terminis, https://en.wikipedia.org/wiki/Contradictio_in_terminis?Antonboat (talk) 12:40, 2 January 2018 (UTC)
No, not at all. Read the definition of a bounded set. The word "size" in the lead of that article means something different from the number of elements in it.--Jasper Deng (talk) 20:00, 2 January 2018 (UTC)

I am very sorry that I overstretched your understanding of English abbreviations with BTW, which is "by the way", and your Dutch association would be, quite analogous, VAT.

I also seem to have used too tedious a formulation in writing that you deny to strictly keep up meanings of notions, even of those you defined yourself (sequence vs. one of its items). The simple reason is that by this you break any conclusive argument you might have had in your intentions. Either you use single elements of this sequence, which are all <1, or you use the whole sequence, which itself has no value, but stands for a value that fits to a certain setting by axioms, which are missing from your statements. Therefore, setting aside all the rubbish from (5) to (10), your claim (11) is rendered meaningless, since no value for 0.(9) has been defined along your way. Math does not allow to simply state the result of a sum of "infinitely many" summands having some property. For e.g., summing 1/n grows beyond all bounds, but summing the reciprocals of arbitrarily small powers (>1) of n does converge, in the sense relevant within reals, and to which you should make yourself acquainted with.

In no way you should assume yourself not to be refuted, just because it is too tedious, as correctly said already, to reply to "insufficiently rigorous" claims.

BTW, you cannot brag with your age considering mine. What does this suggest to you? Purgy (talk) 09:20, 21 December 2017 (UTC)

Dear Purgy. I begin with an answer to your last question. I get the idea that I must say: “thou” instead of “you”, but thou implies: “Thou“. Therefore I leave it so and say “you.” I accept that I’m growing old, but I don’t brag with it. Why should I ? You write: “In no way you should assume yourself not to be refuted, just because it is too tedious, as correctly said already, to reply to "insufficiently rigorous" claims”. When I was young, some years ago, I wanted to reach the unreachable like Don Guijote de la Mancha. But now I’m to old and maybe too unwise for that. Besides I have to sing and play with our grandson who is somewhat more than two years old. But I think it is a pity that I can not introduce an, according to me sound principle as Euclid's Theorem and its consequences, in mathematics because I miss the necessary mathematical knowledge. Dear Purgy what advise can you give me in this matter? — Preceding unsigned comment added by Antonboat (talkcontribs) 10:29, 21 December 2017 (UTC)
For starters I'd start by giving up thinking about mathematics as being a part of physics, that distance in mathematics has a direct link to distance in the real world. Penrose for instance talks about three worlds - the actual world, the world of physical theories, and the world of mathematics, see [1] for a description. And there where they say (the student, not Penrose) "I am not sure if there may be some mathematical truths that are simply beyond human reason, though", we have proved there are parts of mathematics which are inaccessible to us. Dmcq (talk) 11:04, 21 December 2017 (UTC)
Dear Dmcq. In my note I meant: ‘travelling distances’, where fysical distances are expressed in mathematical numbers. You write: “... we have proved there are parts of mathematics which are inaccessible to us.” In that case I think: Be carefull!, but than I become curious too. There are lots of things which are inaccessable to us, for example: to know what is the essence of electricity. But the electrical laws can be excellently expressed in mathematical terms. So if there are parts in mathematics inaccessable to any mathematician, I think there can be reason to distrust, because I think there is a big chance then that these parts are as illusive as the speaking wolf in Little Red Riding Hood.Antonboat (talk) 10:03, 22 December 2017 (UTC)
A good example of what I was trying to say. You are mixing up your conception of the physical world with actual physics and you are mixing that up again with mathematics. In current physics real numbers are sill used to model distances but we're not a all certain it means much at very small distances because of limitations from quantum mechanics. Our physical conception has changed because of experimental data. The mathematical idea of the standard real number system is unaffected by this and does not intrinsically have anything to do with physical distance, it just was based on the original naive conception of distance in the real world but actually is now based on axioms. Dmcq (talk) 14:09, 22 December 2017 (UTC)
Dear Dmcq. In an effort to prevent further discussion on ’distances’ the following. Jasper deng, Jasper Deng (talk) 17:46, 21 December 2017 (UTC), wrote to me: “While it is true that no term of the sequence is exactly 1, its limit is 1 and we define the notation 0.(9) to denote the limit of the sequence, not any particular term.” I answered to this: That’s what I meant with my suggestion: N.B.1: I suggest to rewrite the notation 0.999... = 1, as in agreement with the Archimedian interpretation of numbers as distances: 0.999... ≠ 1 nearing but not reaching a limit 1. “ That would, in my opinion, prevent a lot of comments to the Argument page in the future. (I’ve deleted the words : “in agreement with the Archimedian interpretation of numbers as distances“ here because they are not relevant for the definition of 0.999....)
Jasper deng wrote me about distances: ... in the physical world one probably cannot meaningfully talk about "distance" shorter than the Planck length, ... ” My response to it was: In: https://en.wikipedia.org/wiki/Planck_length, is said “Much like the rest of the Planck units, there is currently no proven physical significance of the Planck length.” If this unit is real and > 0 it can be divided. If it’s pure hypothetical it can be illusive, and such units you can only divide by illusive dividing.
I wish you a Merry Christmas.Antonboat (talk) 16:31, 22 December 2017 (UTC)
I was trying to answer " I miss the necessary mathematical knowledge. Dear Purgy what advise can you give me in this matter?" Dmcq (talk) 18:28, 22 December 2017 (UTC)
@Antonboat: You can get a real number by dividing the Planck length by something greater than 1, but the hypothesis is that there can exist no two objects whose distance is that, because two objects that close don't have a well-defined distance.--Jasper Deng (talk) 20:04, 22 December 2017 (UTC)

Yes, merry Christmas to all, who believe in Cantor's diagonal, but also to intuitionists, constructivists, (ultra-)finitists, and also to continuous classical and discrete quantum- and string-, and loop-gravity-physicists, to all Hausdorffians and non-compacts, in wild spaces, to those in Minkowski spaces or Calabi-Yau-manifolds.

I hope math can survive that when 0.(9) < 1, then also

and :)


I'll light the 4-th candle on my Advent wreath, and, even when ungendered: "Peace on earth, and good will to men". Purgy (talk) 17:50, 22 December 2017 (UTC)

Dear Purgy. Thank you for your good wishes. There can be miracles when you believe. (The Prince of Egypt.)Antonboat (talk) 12:51, 2 January 2018 (UTC)
A Christmas cracker joke from Marcus du Sautoy
Teacher: Today we're learning about large numbers. What does anyone think is the largest number?
A pupil: Seventy three million five thousand and twelve.
Teacher: How about Seventy three million five thousand and thirteen?
Pupil: Well, at least I was close!
 :) Dmcq (talk) 23:52, 24 December 2017 (UTC)
Dear Dmcq. Both are wrong. It’s seventy three million five thousand and twelve and a half.Antonboat (talk) 13:02, 2 January 2018 (UTC)

Why the infinite multitude of nines of 0.999... can’t have an infinite number.[edit]

I think I found the following fundamental ‘soft spots’ in Cantor’s theory of infinity:

1) He didn’t proof that the set {0, 1, 2, 3, etc.} could be an actual infinite set: a set that really could have all the natural numbers, none excepted. (According to: https://en.wikipedia.org/wiki/Axiom_of_infinity, sub: “Interpretation and consequences”, the existence of such a set is even now not unambigously demonstrated.) Euclid’s Theorem, https://en.wikipedia.org/wiki/Euclid%27s_theorem, proves that there cannot be a largest prime. As primes are natural numbers, so there cannot be a largest natural too. And if there can’t be a largest natural, the only actual possible infinite set of naturals must be an unbounded set of naturals, that, being unbounded, can impossibly actually have all the naturals, none excepted.
2) He didn’t define the infinite numbers in the sense that he didn’t indicate the exact spot on a number line where, as successor of a natural, not a natural but an infinite number could appear on it. (Even now, such an unambiguous definition of an infinite number is still not given in: https://en.wikipedia.org/wiki/Aleph_number, sub: Aleph number.) But Cantor was convinced that infinite numbers could really exist because he thought he had proved this his diagonal argument. But can they really exist?
3) The conclusion he came to by interpreting the result he found in his diagonal argument can impossibly be true.
On the site: https://en.wikipedia.org/wiki/Cantor%27s_diagonal_argument, in: “Uncountable set” is written: “In his 1891 article, Cantor considered the set T of all infinite sequences of binary digits (i.e. each digit is zero or one). He begins with a constructive proof of the following theorem: If , , … , , … is any enumeration of elements from T, then there is always an element of T which corresponds to no in the enumeration. By construction, differs from each , since their th digits differ (highlighted in the example). ((As I expect you to be familiar with Cantor’s way of working, I omitted that example here, Antonboat)) Hence, cannot occur in the enumeration. Based on this theorem, Cantor then uses a proof by contradiction to show that: The set T is uncountable. He assumes for contradiction that T was countable. Then all its elements could be written as an enumeration , , … , , …. Applying the previous theorem to this enumeration would produce a sequence not belonging to the enumeration. However, was an element of T and should therefore be in the enumeration. This contradicts the original assumption, so T must be uncountable.“
But is this conclusion true? In my opinion it cannot be true. And that can easily be shown. One may write the following paraphrase of the proof above in digits, instead of the binary digits Cantor used. Suppose that an infinite list could be made in which each positive integer is matched up with, in random order, a real number between 0 and 1, the reals given by infinite decimals. The first five lines of this list might look then as follows:
0.44159...
0.34333...
0.71428...
0.41441...
0.50004...
... ...
The digits that run down the diagonal in underlined boldface are: 4, 4, 4, 4, 4, …. By subtracting 1 from these digits, a number: 0.33333... is found. This number is not in the here given list of real numbers.
Sub: ‘Interpretation’ in Wikipedia Cantor's diagonal argument, https://en.wikipedia.org/wiki/Cantor%27s_diagonal_argument, is said: “In the context of classical mathematics, … the diagonal argument establishes that, although both sets are infinite, there are actually more infinite sequences of ones and zeros (read here for ones and zeros: digits, Antonboat) than there are natural numbers.”
But can this interpretation be right indeed? Every mathematician knows that an actual infinite list of reals < 1 has to contain reals like: 0.44444…, 0.33333…, 0.999…, 0.5, 0.9…, etc. The list of reals < 1 above is interpreted as an infinite list of all the reals < 1, none excepted. But looking at it we see in reality a list containing only 5 elements, thus a list in which the group of the still remaining infinite multitude of possible reals < 1 not is included. That’s the reason that the list of reals < 1 above, can impossible be an actual infinite enumeration of the reals < 1. And that’s why the in the classical mathematics conventional interpretation of the result of this argument: that it proofs that there are actually more reals < 1 than there are natural numbers, can’t be true. In my opinion the result found by means of the diagonal argument, must be interpreted as a definite proof that the reals < 1 impossibly could be enumerated in an actual infinite and complete list. In that way the diagonal argument proofs for the reals < 1, what Euclid’s Theorem proved for the primes and thus the naturals.

From these proofs together follows that the assumption of the existence of an actual infinite set, be it naturals or reals, limited by ‘infinite’, must be illogical. And if there can’t be a limit ‘infinite’ there can’t be infinite numbers. (In that case the reals < 1, enumerated in random order: 0.99, 0.9, 0.9999, 0.999, etc, can boundlessly be paired with the enumeration: 0, 1, 2, 3, etc, in this way: [0, 0.99], [1, 0.9], [2, 0.9999], [3, 0.999, [etc, etc].) And the nine in the notation of the repeating nine: 0.(9) must then be interpreted as a bounded series of one nine, to which boundlessly is added a new nine. This infinite process of boundlessly adding can’t have a limit ‘infinite’ because of its boundlessness. So any possible number of nines, (and naturals etc.) can only be expressed in a natural number. The only way that remains then to interpret and define ‘infinite’ could be: defining ‘infinite’ as a process of unbounded increase or decrease (in number or length), or as a process of boundlessly continuing, (of time).Antonboat (talk) 13:12, 2 January 2018 (UTC)

A good and happy 2018 to you all!˜˜˜˜ — Preceding unsigned comment added by Antonboat (talkcontribs) 10:32, 2 January 2018 (UTC)
@Antonboat: Wrong. You must start with fundamentals like the rigorous definition of an infinite set. The rest of your post is the same hand-waving that got you nowhere before Christmas. If you are rejecting the axiom of infinity then you cannot possibly be talking about the natural numbers, an object whose existence is the axiom of infinity.--Jasper Deng (talk) 10:38, 2 January 2018 (UTC)
Dear Jasper Deng You write: ”If you are rejecting the axiom of infinity then you cannot possibly be talking about the natural numbers, an object whose existence is the axiom of infinity.” Before there was an axiom of infinity there was Euclides and thus there were primes/ natural numbers.Antonboat (talk) 13:07, 2 January 2018 (UTC)
@Antonboat: What came before the axiomization of mathematics is not sufficiently rigorous. Why are you so resistant to embracing the rigor I expect of you for a discussion like this?--Jasper Deng (talk) 18:08, 2 January 2018 (UTC)
Dear Jasper Deng. You told me I had to be more rigorous in my reasoning. So I tried this in this note by citing Wikipedia. Apparently not to your full satisfaction. But is your comment a substantive comment? I’m afraid it is not. In my opinion, you’re barking again without biting. An axiom is as far as I know a hypothesis and not a rigorous proof.
Show to me by a rigourous proof that there can be a infinite set of all the naturals, none excepted.
Show me the exact spot on a number line where naturals change in infinite numbers.
Show me that I’m wrong in my expecting that reals like 0.3333, 0.9 and 0.314 should be in in a set of which is assumed that it can be the set of all reals< 1, none excepted.
Assuming you to be an able mathematician, I expect from you brainy answers, not any rudeness. Or am I asking too much? In that case please: "Sit!" and: "Be still!".Antonboat (talk) 07:19, 4 January 2018 (UTC)
I already told you: the existence of the natural numbers as an infinite set is a fundamental axiom of ZFC set theory and as such, cannot be proven from the other axioms of that theory; in fact, the definition of "finite" and "infinite" relies on the existence of that set so this is pretty much by definition. To prove it requires assuming something at least as strong as that. Merely "citing" Wikipedia does not count as "rigor". Refer to Diego's comments on this page for what I mean by it. And there is no "exact spot on a number line where naturals change in infinite numbers", because the infinite ordinals are not contained in the natural numbers and a number line represents only the set of natural numbers.--Jasper Deng (talk) 07:23, 4 January 2018 (UTC)
(edit conflict)Jasper, please don't bite the newbies. If Antonboat does not have a formal knowledge of the axiomatization of real numbers, you should participate in the discussion by explaining how he can get it and pointing out some good books he can read to acquire it, not complaining that he doesn't use a theory that he doesn't know about. Diego (talk) 23:09, 2 January 2018 (UTC)
Dear Diego.Thank you for your understanding of my position in this discussion. But I've learned to accept real truths although they may be hard to swallow. For, although I may be a newby in your eyes, I'm absolute not a baby. I want to know as we say in Dutch: "het naadje van de kous" (literally translated: "the seam of the stocking"), by which I mean that I should like to know the smallest fundamental details of something, in this case the rigorous proofs, not axiomata!, a mathematician could give me as an answer to the questions I asked Jasper Deng above.Antonboat (talk) 07:51, 4 January 2018 (UTC)
@Diego Moya: I don't think I need to be any less blunt when I have pointed him to in the right direction something like a dozen times above, including links to the relevant articles, and detailed explanations of how to approach this topic. I'm emphatically not an intuitionist and usually don't believe that's the best pedagogical approach. I also do not believe that providing him with a list of books (or other offline resource) is going to be very useful for him, as he would have to go out of his way to obtain those books.--Jasper Deng (talk) 08:11, 3 January 2018 (UTC)
That argument reminds me a bit of Giovanni Girolamo Saccheri saying when trying to prove the parallel postulate "the hypothesis of the acute angle is absolutely false; because it is repugnant to the nature of straight lines". And Saccheri actually did know some mathematics. The modern requirements of rigor have meant that Euclid's axioms have had to be augmented with a whole load of new ones - you might like to have a look at Hilbert's axioms and then Tarski's axioms to see what is meant by that. In them one could replace the word point by quib and line by walb etc and the arguments would still hold and give valid theorems. The natural numbers and infinity are like that in modern mathematics. Dmcq (talk) 23:06, 2 January 2018 (UTC)

Hello Antonboat. "Newbie" is a term we use for new editors of Wikipedia; it speaks only about your exposition to the community norms and customs, not your life experience. Your account information tells me that you've made just 48 edits in total starting last december, which makes you a newcomer in this sense.

From your words I think you have a confusion regarding the role of axioms and hypotheses in logic and maths; both are logic assertions, but they serve different purposes. I'll try to provide a very brief explanation; it's possible that you already have some knowledge about the following, but I think this will help explain a misscomunication that's going on in this conversation.

  • In formal logic, a theory is a well-defined set of (umambiguous) logic assertions called axioms, plus all the logic assertions ("theorems") that can derived from that set using inference rules.
-If one of the axioms in the original set can be derived from the others, you can remove it from the set and get an equivalent but simpler theory. So in practice, you can assume that all the axioms in a theory are independent from each other, and all are needed to define the theory.
- If you add a new independent axiom to the original set, you get a new theory, different from the previous one, which may have different logical consequences.
  • A theory is consistent if it is impossible to derive a contradiction from the set of axioms using the allowed inference rules.
  • A hypothesis is a logic assertion that you are trying to derive from the set of axioms in the theory. If you can find that derivation, the hypothesis is proven and becomes a theorem. But if you create a theory containing the axioms and the hypothesis, and you can derive a contradiction from that, the hypothesis is disproven.

In this conversation, you're treating the existence of infinite sets as a hypothesis that needs to be proved (or disproved) from the axioms of natural and real numbers. However, this happens to be an unreasonable request, since it is known that it is impossible to create such proof from those axioms; and also that it is impossible to disprove the existence of infinite sets from them: the theory containing the axioms of the reals plus the existence of infinite sets is consistent. (By the way, your attempt at a disproof is not precise enough to count as a formal logic deduction). This existence or inexistence of infinite sets is not a theorem in the theory of naturals and reals, i.e. not an assertion that can be derived from their axioms. So, when mathematicians use infinite sets, they do it as a new axiom, because their existence is an independent logic assertion, and therefore it can only be treated as an independent axiom that expands the basic number theory.

I hope this helps. I'm sure the other editors will correct me if I've made some mistake or over-simplification somewhere. ;-) Diego (talk) 10:08, 4 January 2018 (UTC)

Dear Diego Moya. I hope you didn’t think I felt offended by you naming me “newby”. Yes I can’t deny it, I’m a newcomer in your world. Thank you for your detailed explanation of the different use of the term axiom in logic and mathematics. But no, I don’t have a confusion regarding the role of axioms and hypotheses in logic and maths. What I miss in your overview is the description of the qualitative difference that exists between both types of axiom. In: https://en.wikipedia.org/wiki/Axiom, sub Axiom, is written: “ Logical axioms are usually statements that are taken to be true within the system of logic they define ... . ... non-logical axioms (e.g., a + b = b + a) are actually substantive assertions about the elements of the domain of a specific mathematical theory (such as arithmetic). When used in the latter sense, "axiom", "postulate", and "assumption" may be used interchangeably. In general, a non-logical axiom is not a self-evident truth, but rather a formal logical expression used in deduction to build a mathematical theory.” So a non-logical axiom like the axiom of infinity, is an assumption like a hypothesis.
You write: “In this conversation, you're treating the existence of infinite sets as a hypothesis that needs to be proved (or disproved) from the axioms of natural and real numbers. “ Yes, you’re right. And one rigorous proven axiom in your mathematics is Euclid’s Theorem that concludes that there can’t be a largest prime and following him I conclude thereafter: and thus no largest natural. You write of this reasoning: “(By the way, your attempt at a disproof is not precise enough to count as a formal logic deduction).” Is the fact that every n + 1 > n not proof enough that there can’t be a set/list, or whatever name a mathematician may give to it, of the infinite multitude of natural numbers, non excepted? Why than accept that Cantor proofs in his diagonal argument that a list/set of all the reals < 1 is impossible?
And especially: why than accept the axiom of infinity as the real truth, knowing Euclid’s Theorem?Antonboat (talk) 09:29, 8 January 2018 (UTC)
@Antonboat: Why can't a set be of infinite cardinality?--Jasper Deng (talk) 09:37, 8 January 2018 (UTC)
Thank you, Diego Moya, for this nice summing up, I truly hope that this covers a good deal of the accumulating misunderstandings. There is just one detail I am a bit weary of. To my knowledge there are several axiomatic systems to specifically establish "natural numbers and their arithmetic", even in slightly different expressive power, but all of them contain an axiom for the existence of the "countable" infinity (induction?). I am not sufficiently educated to be similarly explicit about the various axiomatic systems for "real numbers" and for "sets" in the various settings (ZF(C), Gödel, ..., transfinite induction?), that all do allow for higher infinities, and that all are capable of implementing the natural numbers.
Besides this, one should also note, that extracting countable lists from the real numbers, to show that there are only countably many such numbers, is begging the question, and that Cantors 'list', spreading infinitely in two directions, which is explicitly conceived and assumed to exhaust all real numbers, is shown to fail this task by not containing at least(!) one other real number, constructed as different to all the countably infinitely many numbers in the list, thereby concluding the proof by contradiction (reductio ad absurdum) of the non-existence of such an exhaustive and countable list. (Remark: this diagonalization argument is somewhat different to the diagonalization argument of proving the countability of rational numbers, also involving a 2-dim list)
Perhaps, it helps to mention additionally that there exists a (small?) fraction of mathematicians, that does not accept these axiomatic systems as a valid basis of mathematical work, some of them additionally discarding parts of the rules for the mentioned derivation of theorems from the axioms, which the vast majority of mathematicians gracefully embrace. Purgy (talk) 11:08, 4 January 2018 (UTC)

How to number any boundless and well ordered enumeration of reals smaller and bigger than one with naturals.[edit]

How could these reals be numbered, presented in a well ordered way? For the naturals this is simple because there is a smallest natural. That’s why numbering the naturals of the unbounded well ordered enumeration of naturals: 0, 1, 2, 3, etc with ordinals 0, 1, 2, 3, etc., is not difficult: pairing both enumerations this way:

[0, 0], [1, 1], [2, 2], [3, 3], [etc, etc].

But a well ordered unbounded enumeration starting from the smallest real, is impossible. The reason for this is that for every real number 10E-n we choose as starting point for an enumeration like: 1.10E-n, 2.10E-n, 3.10E-n, we could choose an infinitely smaller one. Where to begin with numbering a well ordered unbounded series of reals? Starting from 1 = 1.10E0 as the supposed smallest real, reals could be paired with ordinals in this way:

[1.10E0, 1], [2.10E0, 2], [3.10E0, 3], [etc, etc].

Starting from 1.10E-1 as the supposed smallest number the pairing:

[1.10E-1, 1], [2.10E-1, 2], [3.10E-1, 3], [etc, etc] is found.

And in general, starting from any 1.10E-n as the supposed smallest real the pairing :

[1.10E-n, 1], [2.10E-n, 2], [3.10E-n, 3], [etc, etc] is found.

For any n applies: if one assigns the ordinal 1 to a real 10E-n, all the reals in the unbounded well ordered enumeration of reals: 1.10E-n, 2.10E-n, 3.10E-n, etc, could be paired to a unique ordinal which can be found by multiplying each real with 10En. As an example the following.The repeating nine 0.(9) may be seen as represented by the unbounded addition: 0.9 + 0.09 + 0.009 + 0.0009, etc. As the first element of this addition is a real: 9.10E-1, the nth element must be a real 9.10E-n. Starting from 10E-4 as the supposed smallest real, 0.0009 must be paired to the ordinal: 0.0009 x 9. 10E4 = 9, 0.009 must be paired to the ordinal 0.009 x 10E4 = 90, 0.09 must be paired to the ordinal: 0.09 x 10E4 = 900 and 0.9 must than be paired to: 0.9 x 10E4 = 9000. Applying this rule then to the real 9999 this real/natural must be numbered with 99999.10E4 = 99,990,000. The only exception to this rule is 0. If 0 is seen as the smallest real, applying this rule leads to the result that only an ordinal 1 could be assigned as shown here: [1.0 = 0, 1], [2.0 = 0, 1], 3.0 = 0, 1], [etc, etc].Antonboat (talk) 06:54, 4 January 2018 (UTC)

You are running into the motivation for having the axiom of choice, which has been needed to show the existence of a well-ordering of the reals. There is no smallest real number, however, just like there is no biggest real number. Be careful with trying to envision infinite sets as infinite "lists". For the real numbers, such a listing does not exist when the indexing set is the natural numbers.--Jasper Deng (talk) 07:11, 4 January 2018 (UTC)
Dear Jasper Deng. You write: “You are running into the motivation for having the axiom of choice, which has been needed to show the existence of a well-ordering of the reals.” Why? In: https://en.wikipedia.org/wiki/Well-orderIn mathematics, is said: “a well-order (or well-ordering or well-order relation) on a set S is a total order on S with the property that every non-empty subset of S has a least element in this ordering. Following this definition I stated that: 1.10E-1, 2.10E-1, 3.10E-1, etc, is a well ordered enumeration. What’s wrong with that? You write: “Be careful with trying to envision infinite sets as infinite "lists". For the real numbers, such a listing does not exist when the indexing set is the natural numbers.” By “infinite list/enumeration” I meant: a bounded list of reals to which bounlessly are added new unique real numbers being multiple negative powers of ten. Following my interpretation of infinite it can be done, for example: [1.10E-1, 1], [2.10E-1, 2], [3.10E-1, 3], [etc, etc].Antonboat (talk) 10:09, 8 January 2018 (UTC)
@Antonboat: No, that is not a well-ordering of all the real numbers. Where does a number like 1/9 = 0.11111111..., pi, or even 0.125 fit in your scheme? Again, you are doing yourself no favors by thinking of infinite sets as "a bounded list of reals to which boundlessly are added new unique real numbers". When you say a "list" or "listing", you must identify an index set in order to make a meaningful (rigorous) statement.--Jasper Deng (talk) 10:16, 8 January 2018 (UTC)
Dear Jasper Deng. It seems you comments faster that I can respond to it. You write: “@Antonboat: No, that is not a well-ordering of all the real numbers. “ You’re right, but, like you, I believe in Cantor’s diagonal argument which proved that you can’t have all the reals, none excepted, in a set/list/enumeration. And in my note I didn;’t pretend that. What I said there was: “For any n applies: if one assigns the ordinal 1 to a real 10E-n, all the reals in the unbounded well ordered enumeration of reals: 1.10E-n, 2.10E-n, 3.10E-n, etc, could be paired to a unique ordinal which can be found by multiplying each real with 10En.” You ask me: “Where does a number like 1/9 = 0.11111111..., pi, or even 0.125 fit in your scheme?” Please see about this my answer to Dmcq, Antonboat (talk) 10:37, 8 January 2018 (UTC) You write: “Again, you are doing yourself no favors by thinking of infinite sets as "a bounded list of reals to which boundlessly are added new unique real numbers".” Again: give me a rigorous proof why I may not do that. You write: “When you say a "list" or "listing", you must identify an index set in order to make a meaningful (rigorous) statement.”As far as I know I did this by the index set {1, 2, 3, etc}Antonboat (talk) 11:02, 8 January 2018 (UTC)
A set is not the same as a list, please take out the time to distinguish the two notions. A list, for our purposes, is a surjective function from the index set to the set we are listing, which is not going to be possible when the index set is the natural numbers and the indexed set is the real numbers. I don't need to disprove something that is not a meaningful mathematical statement, as is your "boundlessly are added" notion.--Jasper Deng (talk) 18:13, 8 January 2018 (UTC)
What you are doing is pointless. It is impossible. Even a simple number like 1/3 doesn't fit in your scheme and in general whatever you do it isn't going to work. It has been proved impossible just like squaring the circle using a straight edge and compass, in fact it is more at the level of trying to find a rational number equal to the hypotenuse of a right angle triangles whose two adjacent sides are both 1. Dmcq (talk) 14:37, 4 January 2018 (UTC)
Dear Dmcq. You write: “What you are doing is pointless. It is impossible. Even a simple number like 1/3 doesn't fit in your scheme and in general whatever you do it isn't going to work. It has been proved impossible ... .¨ I’m sorry to say but both a simple number like 1/3 and an difficult number like π fit perfectly in my scheme, witness the following. It's commonly known that neither 1/3 nor π can be assigned an exact value within the decimal system, witness 1 : 3 = 0.3 + 1/30. If it is supposed that 10E-4 is the smallest real, applying the rule given in my note for 1/3 the ordinal 0.3333 x 10,000 = 3333 is found and for π the ordinal 3,1415 x 10,000 = 31415. For 3333 we find than the ordinal: 33,330,000, for 33,330,000 the ordinal: 333,300,000,000, etc. Supposing 10E-n is the smallest real and n being 100,000 we find the ordinal : 10E-100,000 x 10E-100,000 = 1.Antonboat (talk) 10:37, 8 January 2018 (UTC)
@Antonboat: Sorry, but you have not defined a function. There is no "smallest" real, so you are not free to choose one. Even disregarding that, your scheme is not one-to-one: 0.3333444... (4/9 - 1000/9999) and 0.3333... (1/3) get assigned the same natural number for the "smallest real" 10-4 and for every "smallest real" 10-n there always exists two rational (in particular, real) numbers which get assigned the same natural number. Please read the definition of a well-ordering closely.--Jasper Deng (talk) 10:44, 8 January 2018 (UTC)
Dear Jasper Deng. You write: “There is no "smallest" real, so you are not free to choose one.” Who or what forbids me to do that? As a mathematician you feel free to assume an infinite set of naturals, none exempted. You write: “Even disregarding that, your scheme is not one-to-one: 0.3333444... (4/9 - 1000/9999) and 0.3333... (1/3) get assigned the same natural number for the "smallest real" 10-4 ... .” When I choose 10E0 then 3 and π have the same ordinal: 3, but not assuming 10E-1 is the smallest real. Then 3 has the ordinal 30 and π the ordinal 31. If you choose 10E-5 as smallest real 0.3333444... has the ordinal number 33334 and 0.3333... the ordinal number 3333.
You write “... and for every "smallest real" 10-n there always exists two rational (in particular, real) numbers which get assigned the same natural number.0 as the smallest real have the same ordinal. “ Please tell me more about this.Antonboat (talk) 11:34, 8 January 2018 (UTC)Antonboat (talk) 11:39, 8 January 2018 (UTC)
@Antonboat: You can't pick any "smallest real" such that your scheme enumerates even all the rational numbers, let alone all the real numbers. If you choose 10-5 then similarly 0.3333344... and 0.333333... are a contradicting pair. I put "smallest real" in quotes because there exists no such object, so you are talking about a nonexistent object.--Jasper Deng (talk) 18:13, 8 January 2018 (UTC)
Dear Jasper Deng. You write: “I put "smallest real" in quotes because there exists no such object, so you are talking about a nonexistent object.” Zermelo and Fraenkel assumed that there could be an infinite set of all the naturals, incidentally without proving that! So why may I not suppose that 10E-5 is the smallest real in my reasoning, explicite stating therein that this assumption can’t be true? Because I’m not a mathematician? You write: “@Antonboat: You can't pick any "smallest real" such that your scheme enumerates even all the rational numbers, let alone all the real numbers.“ a) My dear Jasper, give me an exact decimal value for 1/3 and I’ll give you an exact natural to number it. b) I never spoke of all the reals, but of all the reals one can create. You write: “If you choose 10-5 then similarly 0.3333344... and 0.333333... are a contradicting pair.“ No my dear Jasper, starting from 10E-5 there can’t be numbers like 0.3333344... and 0.333333... Only a number 0.33333. If you want to pair the exact decimal values 0.3333344 and 0.333333 with natural numbers you’ll have to start numbering from 10E-7 and then find the ordinals 3333344 and 3333333 for them. And of course you know that numbers like 0.3333344... and 0.333333... are not defined in the decimal system. And we’re arguing about the numbering of reals belonging to the decimal system 10E-n to 10En , any n being < n + 1.Antonboat (talk) 08:00, 9 January 2018 (UTC)
@Antonboat: Sorry, this is once again not even wrong. None of your proposed "numberings" covers even an infinite subset of the reals, let alone all of them. A function can assign one and only one value to each member of its domain. So you are not free to keep moving the goalposts by changing your value of n. Doing so produces a different enumeration and is not valid for all the reals - no finite value of n will suffice to cover all real numbers. For what it's worth, the set of real numbers with finite decimal expansion is countable, though not using what you propose (which does not even cover that subset of reals). But almost all real numbers have infinite decimal expansions.--Jasper Deng (talk) 08:05, 9 January 2018 (UTC)
Dear Jasper Deng "Not even wrong" is barking, not biting. And: "Sticks and stones may break my bones but you could never hurt me" You write: “None of your proposed "numberings" covers even an infinite subset of the reals, let alone all of them. “ Again, for I’ve asked you many times before, give me a rigorous proof that an infinite set of all the reals, none excepted is possible. In my comments on Cantor’s diagonal argument I showed that, starting from bounded sets of reals and naturals he concluded that an infinite set of all the reals < 1 could not be possible, because this argument showed the possibility of creating a boundless multitude of reals. You write: “A function can assign one and only one value to each member of its domain.“ you’re right. You write: “So you are not free to keep moving the goalposts by changing your value of n. “ In my opinion You’re wrong here. If I choose a new domain I may choose a new value for n. You write: “Doing so produces a different enumeration and is not valid for all the reals ...” You’re right it’s only valid for the domain that I chose. Please see also about this my new section: Why Cantor couldn’t succeed in his effort to number the infinite multitude of reals with natural numbers¨ You write: “ But almost all real numbers have infinite decimal expansions.”It’s a pity but then they have no exact decimal value and so can’t be numbered by a natural belonging to the decimal system.Antonboat (talk) 09:17, 9 January 2018 (UTC)
@Antonboat: I'm afraid I'm going to have to be frank here, by restating what I said above. If you find yourself unable to grasp fundamentals like the rigorous definition of infinity and how to use rigor in mathematical discussion, we won't have this conversation.--Jasper Deng (talk) 09:21, 9 January 2018 (UTC)

Why Cantor couldn’t succeed in his effort to number the infinite multitude of reals with natural numbers[edit]

Cantor believed he had proved that there could not be naturals left after numbering all the possible reals smaller than one. It is quite understandable that he came to this conclusion. Let’s suppose that 10E-n is the smallest real in the decimal system. Then 10En must be the largest real in this system. Starting from 1.10E-n as the smallest real, we must conclude that the enumeration 1.10E-n, 2.10E-n, 3.10E-n must stop when we reach 10E-n.10E-n = 1. By supposing that 10En is the largest real there can be no naturals left to number the reals ≥ 1. That’s why he had to come to the conclusion that the infinite number of reals had to be infinite larger than the infinite of the reals. From it he deduced: as there are no naturals left to number the real 1 + 10E-n there must be numbers larger than any natural number: infinite numbers. A great find, found by a great and daring explorer. But could he be right in thinking this way? Let us see. Cantor proved in his diagonal argument that starting from a bounded enumeration of reals < 1 there could boundlessly be reals < 1 added to this enumeration. Euclid proved the same for the primes, thus for the naturals, using the principle n + 1 > n. That’s why for every real we suppose to be the smallest real, there is an infinite multitude of reals smaller than that. That’s why for every real/ natural we suppose to be the largest real/natural there is an infinity of reals/naturals larger than that. And in that case we can number the real 1 + 10E-n with a natural: 10En + 1, the real 2 with 2.10En, the real 10nEn with 10nEn.10nEn , the real 1,000,000 with: 1,000,000.10En and so on. And Euclid’s theorem proved for the primes, thus naturals, that it is impossible to exhaust the for this numbering available infinite multitude of naturals. Challenge to the reader: show me by a brainy reasoning that I'm wrong here.Antonboat (talk) 07:23, 9 January 2018 (UTC)

You say: "... {nonsense} ... But could he be right in thinking this way?" Cantor never thought this way. Mathematicians don't think this way Face-smile.svg. - DVdm (talk) 07:47, 9 January 2018 (UTC)
Dear DVdm. a) Thank you for your evaluation of my note. I invite you here to give a brainier than "nonsense". b) Can you prove that Cantor has never thought this way. Best regards.Antonboat (talk) 08:25, 9 January 2018 (UTC)
Nonsense is nonsense. It can't get brainier than "nonsense". Sorry. - DVdm (talk) 08:51, 9 January 2018 (UTC)
@DVdm: I'm wondering if I am being productive attempting to explain this to Antonboat. Obviously, I'm somewhat frustrated that he doesn't want to start with first principles.--Jasper Deng (talk) 08:54, 9 January 2018 (UTC)
@Jasper Deng: it is clear to me that everyone is wasting their time here. And frankly, I'm reluctant to be completely frank... - DVdm (talk) 09:04, 9 January 2018 (UTC)
@DVdm: I mean, I don't want User:Antonboat to feel bad, but I was just wondering if I was the only one feeling that way, since Antonboat is simply not asking the right questions, a prerequisite for learning math.--Jasper Deng (talk) 09:08, 9 January 2018 (UTC)
I don't think we can help him ask the right questions. I think we must say sorry, please go elsewhere, nobody can help you here. But this is getting off-topic (in an off-topic part of Wikipedia to begin with Face-wink.svg) - DVdm (talk) 09:16, 9 January 2018 (UTC)
@DVdm: Beat you to it (see my previous edit).--Jasper Deng (talk) 09:18, 9 January 2018 (UTC)
@Antonboat: "Cantor believed he had proved that there could not be naturals left after numbering all the possible reals smaller than one." - please state exactly what you mean by this in a well-defined manner. To better understand the subtleties of different infinite cardinalities, read Hilbert's paradox of the grand hotel. The rest of your comment is a complete misunderstanding of Cantor's argument.--Jasper Deng (talk) 07:57, 9 January 2018 (UTC)
Dear Jasper Deng. You wrote to me on 09:49, 23 December 2017 (UTC) “The set of real numbers between 0 and 1 exclusive is a bounded subset of the real numbers and yet is of infinite cardinality.”Antonboat (talk) 08:26, 9 January 2018 (UTC)
Okay and how does that relate to what you said?--Jasper Deng (talk) 08:27, 9 January 2018 (UTC)
My, but possibly not Cantor's, way of reasoning is: "By supposing that 10En is the largest real there can be no naturals left to number the reals ≥ 1.¨Antonboat (talk) 08:38, 9 January 2018 (UTC)
@Antonboat: This is not a meaningful statement. What does it mean to have "naturals left"? To "number the reals"? You are technically not working with the real numbers themselves when you claim there is a "largest" real, so I have been understanding you as working with a subset of the reals. But even then, your argument fails and is not even close to what Cantor is saying. What Cantor is saying is that there is no surjective function from the natural numbers to the reals, or even an open interval within the reals - i.e. there is no "numbering" to speak of in the first place. And again, please drop your conception of "boundlessly adding", per what I said above. I would also repeat what Purgy said above -- wean yourself off of decimal notation for real numbers. It seems to be blinding you to the fundamentals of a construction of the real numbers, which makes no appeal to decimal expansions at all.--Jasper Deng (talk) 08:43, 9 January 2018 (UTC)

Thank you to you all[edit]

Dear Dmcq, Double sharp, DVdm, Purgy, Sławomir Biały, Jasper Deng, Trovatore, Diego Moya, let me begin with thanking you all for the patience you had with me: an old and stubborn non-mathematician. I feel realy honored by the trouble you've given yourselves to convince me of, what you saw as my errors concerning mathematical infinity. I am sure you have been surprised (maybe even angry) at my tenacity to stay with these errors, regardless of the effort you have made to change that. Yes Jasper Deng, I find myself unable to grasp fundamentals like the rigorous definition of infinity as an axiom, because no mathematician has ever proved that an actual infinite enumeration/set whatever of the naturals is possible. I know illusions when I’m confronted with it. Such a set must be an illusion because of its property that it consists of all the naturals none excepted .Because of this property it must be bounded. And this property conflicts in my opinion with its supposed property that it is infinite: boundless. Where you discovered that it is very difficult to argue with someone who only believes in what can unequivocally be proven, it was for me very difficult, but very instructive too, I found out, to argue with people who without rigorous proof, believe in the possibility of such a set. It reminds me, no offense intended, of the believers who believe because its absurd what they believe. But: there can be miracles (only?) when you believe. Het ga jullie allen goed!!Antonboat (talk) 10:51, 9 January 2018 (UTC)Antonboat (talk) 12:10, 9 January 2018 (UTC)

Well I think the main problem is you are confusing physics and mathematics and even then making unwarranted assumptions about physics as always being finite because we can only have a finite number of thoughts. There is a small set of people who have ideas along that sort of line, Max Tegmark with his mathematical universe and Doron Zeilberger who thinks there is a maximun natural number are perhaps the best known, and you might also like Rational trigonometry by Normal Wildberger. Even with ideas like those where anything like a real number may be countable in mathematics one can't within any given system have a way of enumerating the reals and this article is about mathematics. Philosophical matters like that are for elsewhere. Dmcq (talk) 13:15, 9 January 2018 (UTC)
Dear Dmcq. I would like to make a few comments about your comments on my farewell, but I won't do it. I agree with Jasper Deng that there is a time to argue and a time to stop with it. You all made it clear to me that my efforts to graft my ideas on your mathematics didn't fall in good earth. Please don't think I 'm disappointed about that. Wikipedia gave me very much more room to express my, according to you all nonsensensical/wrong/ not-mathematical ideas, than I got here in the Netherlands or elsewhere. I'm very thankfull for that, even for the sometimes grumpy comments. It's all in the game.Antonboat (talk) 10:49, 10 January 2018 (UTC)


Simply Put[edit]

Simply put, the premises of this section are not even wrong. This is not worth continuing. The IP is informed that they will get nowhere with vague hand waving like this (they don't even define what rigorous construction of infinitesimals they consider) and that absent a huge amount of reliable sources supporting such an interpretation, we will not entertain inserting this into the article.--Jasper Deng (talk) 07:13, 30 March 2018 (UTC)
The following discussion is closed. Please do not modify it. Subsequent comments should be made on the appropriate discussion page. No further edits should be made to this discussion.

It is true that "Ellipses denote approximations which ignore infinitesimally small remainders".

The infinitesimally small remainder being ignored by those who claim 0.999... is equal to 1, is the smallest positive quantity represented by 0.000...1

On a number line, there are an infinite number of points between 0 and 1. The first point after zero is 0.000...1 and the last point before 1 is 0.999...

0.999... is the largest possible quantity below 1.

0.000...1 is the smallest possible positive quantity.

Therefore, it makes sense that 0.999... + 0.000...1 would equal 1.

Since this is the case, 0.999... is not quite equal to 1. — Preceding unsigned comment added by 75.89.204.103 (talk) 05:09, 28 March 2018 (UTC)

The standard real number system used for all practical purposes does not have infinitesimals. Dmcq (talk) 09:22, 28 March 2018 (UTC)
Simply put ... not a single claim from above holds within the context of real numbers.
  1. The ellipses here have a well defined meaning (see #3), aside from approximations, and -as said- there are no infinitesimally small remainders within the reals, they are intentionally excluded.
  2. There are no infinitesimal small remainders and no smallest positive numbers (see #5) within the reals, and the notation 0.000...1 is not a well defined decimal number.
  3. The point representing the half of a hypothetical 0.000...1 would be between 0 and 0.000...1, so the latter is not a first point. The number denoted by 0.999... is delimited (from below!) to be not less than 0.(9)n for arbitrary n (not before or below 1).
  4. See the second part in #3, it is about not less than all of these, not about <1. Including a desired result in the premises is called "wishful thinking".
  5. There is no "smallest positive quantity" x within the reals, because x/2 is still positive and is smaller (see also the first part of #3).
  6. The second summand does not exist, and therefore, the consequence does not make sense.
Ex falso quodlibet, or EXPLOSION! Sorry, Purgy (talk) 12:18, 28 March 2018 (UTC)


I don't care (nor does reality) whether you call a number real, unreal, or Aunt Polly. There is a first point on a number line after 0.

I have expressed it as 0.000...1 and if you halve that, I contend that it would be expressed the same way.

Therefore, it would still be the first point expressed just as I have stated.

If you subtract that quantity from 1, you get 0.999...

Therefore 1 is not equal to 0.999... and there would still be an infinite number of points between 0.000...1 and 0.999...

To put it another way,

0.000...1 is equal to 1/∞

and 0.999... is equal to 1 - 1/∞

and (1 - 1/∞) + 1/∞ is equal to 1.

Your intentional exclusions are your downfall. — Preceding unsigned comment added by 98.22.230.8 (talk) 04:40, 29 March 2018 (UTC)

OK, have fun with your infinitely many, halved but equal, inexistent objects within your reality, but I see no chance anymore to help you towards math as it is usually employed. Purgy (talk) 06:43, 29 March 2018 (UTC)


Your intentional exclusion of the infinitesimal may be necessary under most circumstances, but at least in this case it causes you to arrive at an erroneous conclusion.

Is the infinitesimal difference intentionally disregarded simply so that the rules of math can be left intact? If math is a science, then you need to step back and look at this as a scientist and be open to the possibility that the rules may need to change in light of this evidence.

You also need to realize that there is a first point on a number line after 0. Otherwise your math does not accurately model reality. Without a first point no thing could ever advance beyond nothing. It would be impossible to pour liquid into a measuring cup because that first infinitesimally small fraction of an atom of liquid could never enter the cup without violating your math rules. Even an eternal journey begins with the first step.

That first point on the number line can be expressed as 1/∞. If you don't like the decimal expression I used, choose your own. It doesn't matter. The last number before 1 is (1- 1/∞) and that would be expressed as 0.999... Those two numbers add up to 1. Because if you bisect the number line between 0 and 1, the two segments add up to 1. We are just bisecting after the first point ( or before the last point if you prefer).

It seems to me that your faith in mathematical rules compels you to adhere to the claim that 1 = 0.999... when faith in the first point on the number line would serve you better.— Preceding unsigned comment added by 98.22.230.8 (talk) 03:58, 30 March 2018 (UTC)

There is no first number on the number line. If your number line has the reals, then this is so by construction. There are other number systems which do have infinitesimals (unlike the reals), but they still don't have a first number. What is half of this first number? No answer you can give will result in any kind of algebraic structure which has the features that you'd expect of one. There's no "faith" in any of this. It's just a matter of picking useful definitions and constructions to do the things you want. And your notion doesn't result in any kind of consistent number system – certainly not not in one that has properties you'd want or expect. You obviously have some interest in mathematics. If that's the case, you'd be better off actually learning about how mathematicians work, and not scoffing at ideas simply because they don't fit with your preconceptions. –Deacon Vorbis (carbon • videos) 04:57, 30 March 2018 (UTC)
I believe they consider the number line to be a physical thing rather than a mathematical concept. And that there is some minimal amount of time called an infinitesimal and our lives are like frames in a film with these minimal size jumps between frames. This has no relation to modern physics even. In Wikipedia terms this is generously called original research. Dmcq (talk) 06:52, 30 March 2018 (UTC)

The discussion above is closed. Please do not modify it. Subsequent comments should be made on the appropriate discussion page. No further edits should be made to this discussion.

Discussions?[edit]

Where's the appropriate discussion page. A link would be useful. — Preceding unsigned comment added by 67.214.17.229 (talk) 23:24, 31 March 2018 (UTC)

WP does not see itself as a discussion board; I think not even its reference desks. This here subpage seems to me, even when questionable and refuted, to be the most appropriate site within WP. Perhaps, you should try stackexchange, or others. Purgy (talk) 09:09, 1 April 2018 (UTC)

Fundamental Misconceptions embodied in article[edit]

I dont know why some people have this 0.999... = 1 as an article of religious belief and become very angry when this is challenged. Current article reflects a very shallow misunderstanding of the topic under discussion.

We must start with basics -- what are integers? 0,1,2, ... this can be intuitively understood. From there we can go to rational numbers -- ratios of integers. Decimals can be defined as fractions with 10^k in the denominator. Algebraic numbers can be defined as roots of polynomials. However THERE IS NO WAY to rigorously DEFINE the symbol 0.999 repeating -- this just cannot be understood intuitively --

All of the existing PROOFS are actually subterfuges. What we are really doing is PRESENTING an intuitively reasonable way of DEFINING this symbol -- which is new and does not correspond directly to anything within the known number systems -- just like the symbol "i" for the imaginary square root of negtive 1 has to be INVENTED and defined and then properties can be assigned to it. Without explicit mention a DEFINITION of the repeating decimal symbol is introduced -- this DEFINITION carries the weight of the proof. There are MANY different ways of DEFINING the repeated decimal fraction all of which can be intuitively justified -- what is not apparent is that there are ALSO definitions which would lead to FAILURE of the equality. So the proof hinges on a HIDDEN DEFINITION.

MOST reasonable definitions of 0.999 repeating will equate this to 1 within a number system which does not have infinitesmals in it. However the decision as to whether or not we allow infinitesmals to exist is just that -- an ARBITRARY decision about how we like to define real numbers. If we allow for the existence of infinitesmals -- which means a number S such that S>0 BUT S < 1/n for all integers n, then 0.999 repeating will be infinitesmally smaller than 1.

This is not something which is subject to PROOF -- it is just a DECISION that we make, as to the set of axioms we would like to use to define our real numbers. To present it as a proof is misleading. Suppose we REPHRASE the question as the following:

DO INFINITESMALS EXIST? There is no answer to this question -- Just like DO IMAGINARY numbers exist? has no answer. We can CHOOSE to answer either YES or NO depending on the purpose for which we are doing the mathematics.

IF infinitesmals DO NOT EXIST (by assumption, at outset) then 0.999repeating will have to equal one because it is easy to show that the difference must be infinitesmal (and indeed, that is what most of the proofs do). Then, since infinitesmals do not exist by assumption, the equality is guaranteed. This essential part of the argument -- that we are assuming in advance - without any justification - that infinitesmals do not exist -- is hidden and not made explicit in the "proof". We cannot prove that infinitesmals do not exist -- we can ONLY assume that they do not -- so in effect all proofs are proofs by assumption. Alternative, the proofs implicitly rule out existence of infinitesmals, without mentioning this, whereas all arguments hinge centrally and crucially on this issue.

IF infinitesmals EXIST -- which we CAN assume, just like we can create a number with or without imaginaries allowed -- then 0.999repeating does not equal 1, RATHER the two will differ by an Infinitesmal amount. So basically the question is WHETHER OR NOT we want to allow infinitesmals into our real number system. This is a DECISION we must make, not a question of PROOF.

Asaduzaman (talk) 06:46, 10 September 2018 (UTC)

I perceive the misconception to be on your side. Maybe the article doesn't sufficiently rub it in for your taste, that it is written in the context of (the well defined, standard) reals, but there are paragraphs within this article, mentioning these other number systems. This article is about the existing and meaningful definition of the string "0.999..." in the context of these reals, and not about the question, whether infinitesimals exist, even when it confirms their existence within the appropriate (other!) number systems. I admit, in a consenting way, that the article does not treat extensively whether this string has a defined meaning elsewhere. There is however no question, whether infinitesimals exist within standard reals: They do not exist there! Neither is there a question about the value of 1 for this string under the submitted definitions. There is neither a subterfuge, nor a hidden definition.
BTW, you yourself and your credentials qualify via your contributions, not the other way round for the latter. Purgy (talk) 11:33, 10 September 2018 (UTC)

Theory and Reality[edit]

There are no infinite objects in our world.
In our world it is impossible to create an infinite object.

The number 0.(9) does not exist and can not exist in reality.
This applies to any infinite number.
Any infinite number is a theory.

If one day we begin to create something infinite in reality, then we will never finish creating it.
Any infinity achieved is not infinity by definition.

Therefore, 0.(9) = 1 only in theory.

When we say that "0.(9) = 1", we mean that the already created number 0.(9) exists, but it exists only in our imagination.


Now consider an example:
x = 1/3 (is always)
x = 0.(3) = 0.33333... (only in theory)
y = 3x (is always)
y = 3 * 1/3 = 3/3 = 1 (is always)
y = 3 * 0.(3) = 3 * 0.33333... = 1 (only in theory)

There is no paradox here. We just did not create an infinite number 0.(3) = 0.33333... in the reality.

When we say:
"x = 1/3 = 0.(3) = 0.33333...", we just deceive ourselves and do not understand it.


One more example:
x = 1/3 = 0.(3) = 0,33333...
y = 3x = 3 * 1/3 = 3 * 0.(3) = 3 * 0.33333...
y = 3x = 1 = 0.(9) = 0.99999...
All this is only a theory.

And how are things really?
x = 1/3 ≠ 0.(3) = 0.33333...
y = 3x = 3 * 1/3 ≠ 3 * 0.(3) = 3 * 0.33333...
y = 3x = 1 ≠ 0.(9) = 0.99999...
0.(9) ≠ 1


I hope that I have completed this eternal argument.

Kirill Dubovitskiy (talk) 03:52, 6 January 2019 (UTC)

Never would I disallow you to personally consider this eternal argument as completed by avoiding the use of decimal representations for non-terminating decimals (e.g.: 0.(3), or 0.33333..., or whatever notation), but in very broad, well informed circles these notations are consistently and fruitfully associated to numbers, the existence of which you evidently do not deny (e.g.: 1/3).
OTOH, you are not given the freedom to simply disallow for the existence of coherent theories, insinuate fallacies, and restrict conceptual realities to certain physical representations.
Please, re-read the article's caveats about the range of "real numbers" addressed in this treatment of your eternal argument. As said, you are free to change the ballpark. Purgy (talk) 09:23, 6 January 2019 (UTC)
"1/3" is a formula, a mechanism, a program, a machine, which is capable of infinitely creating an infinite number: 0.33333333...
Or just 0.(3) is an ordinary short form.
Take a piece of paper and a pen and try to create a number completely: 0.33333... (with an infinite number of "3"), probably then you will understand what is being said.
And also "real numbers" have nothing to do with my evidence.
And yes, we can say that 0.(3) is also an instruction or program for a machine or a machine for the production of an infinite number 0.33333...
But then it turns out that:
1/3 = 0.(3) ≠ 0.33333... or even 1/3 ≠ 0.(3) ≠ 0.33333...
And therefore we simply agree among ourselves that 0.(3) is just a shorter way of writing an infinite number 0.33333...
Kirill Dubovitskiy (talk) 03:52, 8 January 2019 (UTC)


0.(9)n ≠ 1 for any positive integer n, but whether 0.999... = 1 is true depends on the definition of 0.999...[edit]

I happened to know the interesting equation 0.999... = 1 through a video on Youtube. I was curious that why people are discussing this for quite a long time since it looks quite obvious that 0.(9)n ≠ 1.

I have to say that I am not an expert on math. To the best of my knowledge, I am giving the following arguments for the interesting debate on 0.999... = 1 which is intuitively incorrect to me. However, the correctness of this equality really depends on the definition of 0.999....

0.999... or 0.(9)n? The potential issue of two previous proofs of the equation 0.999... = 1.[edit]

Below is a 'proof' of the equation 1 = 0.999...:

The issue of the above 'proof' becomes clear if we write it in another way:


Another well-known 'proof' of the equation 1 = 0.999... is that:

Since 13 = 0.333... (taught in elementary school),
we have 3 × ​13 = 3 × 0.333... (by algebra),
that is 1 = 0.999... (by algebra).

However, as it has been widely pointed out, is 13 = 0.333... correct? It depends on how we define 0.333.... Nevertheless, we can say: 1 divided by 3 equals 0.3 with a remainder of 0.1, which can be written as:

13 = 0.3 R 0.1.

Or similarly, we can say:

13 = 0.33 R 0.01,
13 = 0.333 R 0.001,
13 = 0.3333 R 0.0001.

And generalised as:

13 = 0.(3)n R 0.(0)n-11,

Based on the above, if we agree that 0.(0)n-11 ≠ 0, it would be clear that 13 ≠ 0.(3)n. Also, by algebra, it can be easily derived that:

3 × ​13 = 3 × 0.(3)n + 0.(0)n-11

which is

1 = 0.(9)n + 0.(0)n-11

and hence 0.(9)n ≠ 1.

What is 0.999...?[edit]

As mentioned at the beginning, whether 0.999... = 1 is true depends on the definition of 0.999....

In the wikipedia page of 0.999..., it is said that "0.999... (also written 0.9, among other ways), denotes the repeating decimal consisting of infinitely many 9s after the decimal point (and one 0 before it). " To me, this explicitly make 0.999... as a number (something like 0.(9) n→∞ or 0.(9)), which I believe should be a member of the sequence sequence (0.9, 0.99, 0.999, ...) or the set {0.(9)n | nZ+}. In this case, 0.999... ≠ 1 since we have shown that 0.(9)n ≠ 1.

However, after the above definition in the wikipedia page of 0.999..., it is also said that "This repeating decimal represents the smallest number no less than every decimal number in the sequence (0.9, 0.99, 0.999, ...)." which is equivalent to the definition of the notation 0.999... as the limit of the sequence (0.9, 0.99, 0.999, ...). There should be no doubt that the limit of the sequence (0.9, 0.99, 0.999, ...) is 1. Hence, in this case, since 0.999... is just a notation, there is no problem to say 0.999... = 1 which is the same as to say something like x = 1. — Preceding unsigned comment added by Snowinnov (talkcontribs)


Please, reread the definitions: 0.9 or 0.999... is –within this article– not defined as 0.(9)n, not for any natural number n, and additionally, neither n→∞ nor ∞ are numbers in any contexts referred to within this article, so the notations 0.(9) n→∞ or 0.(9) are not covered by the undisputed proposition "0.(9)n≠1 for any natural number n". There are no objections to "1 = 0.(9)n + 0.(0)n-11 for any natural number n", however taking the limit "n→∞" takes these notations beyond their capabilities and the rigorous application of formally defined limits must take over. The claim that 0.999... is a member of the sequence (0.(9)n)n∈ℕ is not sustainable, because there is no such n, the limit of the sequence is not contained in it. Purgy (talk) 16:53, 19 February 2019 (UTC)


All arguments for equality can be defeated, including the limit argument[edit]

I think it's clear that this discussion is no longer productive, and nothing is served by keeping the thread open. –Deacon Vorbis (carbon • videos) 14:33, 15 March 2020 (UTC)
The following discussion is closed. Please do not modify it. Subsequent comments should be made on the appropriate discussion page. No further edits should be made to this discussion.

One indication of this article's flaws is that the only official argument for 0.999... = 1 is the limit argument. All other arguments/proofs are straw men; they are false arguments that can easily be defeated. For example, the flaw in the formal proof on the '0.999...' Wikipedia page is that it does not allow x to be specified in terms of its sum to the nth term whereas it does allow 0.999... to be specified in terms of the sum to its nth term. We can use the nth sum of x = 1 – 0.5/10n to give us a value where the nth sum is always half way between 0.999… and 1. Here we have:

for any positive integer n.

This simplifies to:

for any positive integer n.

And we can see that this does hold for any positive integer n. This shows that if we allow x to be treated in the same way as 0.999…, then there are an endless amount of ‘numbers’ between 0.999… and 1. Indeed, by considering the nth sum, the only thing we can prove is inequality. The nth sum of 0.(9)n will never equal the nth sum of 1.(0)n and therefore these two cannot be equal.

Now let's consider the official argument. Limits and convergence were introduced in the early 19th century but they have always had their critics. In the case of 0.999..., the sequence is 0.9, 0.99, 0.999, and so on, and the limit of this sequence is said to be 1. But this sequence is not a finite structure that is continually being extended; it supposedly preexists as a static abstract object containing 'infinitely many' terms where each term corresponds to a digit in the infinite decimal 0.999... In other words, the limit argument requires that an actual infinity of terms must be possible.

But this type of actual infinity has had counter-arguments going back to the Ancient Greeks over 2,000 years ago. For example, consider a continuous abstract line of length 2 units. If an actual infinity of parts were possible, then the first 1 unit of this 2 units length should be able to exist as the infinitely many lengths 9/10 + 9/100 + 9/1000 + ... corresponding to 0.999... But this causes at least two contradictions. Since the whole line is continuous, then there must be a 'last part' of the infinitely many lengths that is connected to the training length of 1. This contradicts the concept of 'infinitely many' which requires there to be no last part. Also since all parts are connected, if we were to count the lengths, then somewhere the count will need to go from a finite value to an infinite one.

And if the limit argument is flawed (because of its reliance on the validity of an actual infinity of terms), then no valid arguments remain for 0.999... equals 1. PenyKarma (talk) 16:59, 9 May 2019 (UTC)

You're kind of missing the point. "0.999..." is a string of symbols (on a page, or a computer display, whatever). "1" is also a string of symbols. Under the real number system, these two strings represent the same number, as do "57 / 57", "4 - 3", and so on. Your analogies with lengths and lines are completely irrelevant, because that's not how the real numbers are defined. Your issue with "actual infinity" (which I find to be a meaningless concept anyway, but that's another story), is also irrelevant, because the set-theoretic frameworks under which the real numbers are defined happily handle infinite sets. And on a side note, saying that something "has had its critics" is empty rhetoric. The idea that the Earth goes around the sun has had its critics as well, but that hardly stops the rest of us from accepting reality. –Deacon Vorbis (carbon • videos) 15:30, 7 May 2019 (UTC)
My claim is that a static version (involving no motion or passage of time) of Zeno's most famous paradox invalidates the limit argument for 0.999... equals 1. You claim that this is completely irrelevant because "that's not how the real numbers are defined". You then give a hint about how you think the real numbers are defined by saying "the set-theoretic frameworks under which the real numbers are defined happily handle infinite sets".
My argument addresses the popular approach to defining real numbers as ‘an equivalence class of rational Cauchy sequences’. In other words, a real number is defined as a container of infinitely many sequences, each of which is infinitely long, and where the difference between any two sequences will be a sequence that tends towards zero. Any sequence corresponding to a so-called 'infinite decimal' (such as 0.9, 0.99, 0.999, etc) will be a Cauchy sequence because its elements become arbitrarily close to each other as the sequence progresses.
In other words, you claim that any counter-augment to the definition of real numbers as an equivalence class of Cauchy sequences is irrelevant because you know of some other definition that uses set theory (and which no doubt relies on the axiom of infinity). To use your own analogy, this is like saying we can reject any proof that the Earth goes around the sun if we have our own axiomatic system where one of our axioms says that the world does not go around the sun.
Currently the Wikipedia page for 'real number' says: "The current standard axiomatic definition is that real numbers form the unique Dedekind-complete ordered field (R ; + ; · ; <), up to an isomorphism,[a] whereas popular constructive definitions of real numbers include declaring them as equivalence classes of Cauchy sequences of rational numbers, Dedekind cuts, or infinite decimal representations, together with precise interpretations for the arithmetic operations and the order relation. All these definitions satisfy the axiomatic definition and are thus equivalent."
I read this as saying that the different definitions are equivalent. So if one of them is invalid then all of them are. Therefore you cannot dismiss a flaw in one of them simply because it is more difficult to locate the equivalent flaw in another one of them.
For pi to exist on the number line, then the number line from zero to pi must be able to be considered as being a series of lines of length 3, 0.1, 0.04, and so on. Furthermore, the line from pi to the point 4 must also be able to be considered as a line on the continuous number line. And since this 'pi to 4' line must be connected to a line immediately before it, then there must be a 'last part' in the series of infinitely many lines from zero to pi mentioned earlier.
I also notice that you have not commented at all on my refutation of the first formal proof of 0.999... equals 1. It is a very simple proof and in my opinion, its flaw is very easy to expose. Do you still agree with the formal proof or do you admit that it is flawed?
Your first point is that I am missing the point. You proceed to argue that the strings of symbols "57 / 57" and "4 - 3" represent the same number. This is Platonism and I reject Platonism. I interpret your first example as a ratio, not a division operation. I also consider it to have a generic real-world meaning such as 57 of something are in one category as compared to 57 of something in a different category. For example, you have 57 apples and I have 57 apples. If we replace "57 / 57" by 1 then it tells us nothing about how many apples each of us have. Similarly your second example might relate to the action of taking 3 apples off a table that originally contained 4.
I believe that mathematics is the use of symbols as shorthand for generic descriptions of the laws of physics. Most mathematicians are Platonists that believe mathematics is something beyond all physics. At this point any condescending mathematician would respond to me by telling me that I don't understand mathematics and I need to go away and read up on it so that I will eventually know better!
All the arguments for 0.999... equals 1 are flawed...
A common argument is that since 1/3 = 0.333… then we can simply multiply both sides by 3 to get 1 = 0.999… This argument requires that we start by accepting that 1/3 equals 0.333… But we cannot start by assuming a rational can equal a repeating decimal because this is precisely what we need to prove.
When we do short/long division for 1 ÷ 3 we follow an algorithm that repeats. We soon see that the trend is a longer (but finite) number of decimal places and a smaller (but always non-zero) remainder. So the long-term trend is a very long decimal and a very small non-zero remainder. The long-term trend is not ‘infinitely many’ digits with a zero remainder.
If we think of 0.333… as 3/10 + 3/100 + 3/1000 + … then the sum up to the nth term is 1 / 3 – 1 / 3(10n) and so this is less than 1/3 for all n. This means that the nth sum is a non-zero distance away from 1/3. This holds for ALL of the terms in 0.333… Since no term can possibly exist where 1/3 is reached, and since 0.333… is nothing more than its terms, it cannot equal 1/3.
Then there is the argument that if we subtract 0.999… from 1 we get zero. If we say 0.999… is the series that has an nth sum of 1 – 1/10n, and 1 is the series that has the nth sum of 1 – 0n then when we subtract 0.999… from 1 we get the series that has an nth sum of 0 + 1/10n
If a series like 0.999… is a valid number, then this answer is equally a valid number. We cannot assert that this result must be numerically equal to 0, because that would mean that our starting position is that 0.999… already equals 1.
Then there is the so-called algebraic proof. We start with x = 0.999... then we multiply both sides by 10 and subtract what we started with to apparently get 9x = 9 thus proving x = 1.
The trick used to pull off this illusion is to misalign the series and then to claim that all trailing terms will cancel out, as shown here:
10x = 90/10 + 90/100 + 90/1000 + …
x = 9/10 + 9/100 + 9/1000 + …
The trick is the misalignment of the terms (terms in the ‘x =‘ line above are shifted 1 place to the right). Such misalignment is invalid because if it was valid we could prove 0=1 by taking 1+1+1+… away from itself (try it yourself). If we align the series correctly then we get this result:
10x — x = 81/10 + 81/100 + 81/1000 + …
Another way to appreciate why the misalignment is invalid is to think of 0.999… as the series 9/10 + 9/100 + 9/1000 + … If we multiply this series by a factor of ten then we don’t change the number of terms; we have the same terms (in terms of one-for-one correspondence) as we started with, only now each term is ten times its original value.
The subtraction 9.999… — 0.999… cannot cancel out all the trailing terms unless this one-to-one relationship (between the original and the multiplied series) is somehow broken, and we get an extra term out of nowhere.
Yet another way to show that this algebraic proof is invalid is to consider the general formula for a geometric series, G, with first term ‘a’ and common ratio ‘r’ (since 0.999… is the geometric series with a=0.9 and r=0.1). If we assume that all matching terms cancel out (to ‘infinity’), then the result of the subtraction simplifies to:
(1/r — 1)G = a/r
Substituting a=0.9 and r=0.1, G=x, this evaluates to '9x = 9'.
The important question is were we correct to assume that the trailing terms canel out all the way to infinity? Well, the resulting expression (above) should apply to all geometric series, both converging and diverging, because none of the manipulations used have any reliance on the values of the variables. So if we can find any values for the variables ‘a’ and ‘r’ where the above statement forms a contradiction, then we will have shown our assumption that all trailing terms cancel out was a mistake.
The values a=1 and r=1 make the above statement evaluate to 0 = 1 and so the algebraic proof for 0.999… = 1 must be invalid.
And since the limit argument is defeated by the static variation of Zeno's most famous paradox, we don't have a single valid argument for 0.999... equals 1. PenyKarma (talk) 16:59, 9 May 2019 (UTC)
Please indent your replies and avoid adding extra blank lines between paragraph. I've fixed your last post up, but see Help:Talk for info on how to use talk pages, thanks. Also, please try to add in a single post rather than a little bit at a time (you can use the preview button if you need). This helps prevent edit conflicts.Deacon Vorbis (carbon • videos) 17:07, 9 May 2019 (UTC)
You said: "I believe that mathematics is the use of symbols as shorthand for generic descriptions of the laws of physics. Most mathematicians are Platonists that believe mathematics is something beyond all physics." I don't like labels like "Platonist". I certainly don't really consider myself one. Also, what I described is more like Formalism, not Platonism. And it's not even that; it's just really pointing out the distinction between the string "57" and the real number it represents, which is not a string of symbols, but rather an equivalence class of Cauchy sequences of rational numbers (or pick your favorite other construction; it doesn't really matter which).
In any case, our personal beliefs are only useful insofar as they provide guidance on what foundational axioms we're likely to work with. If you have some sort of personal problem with standard set theory, and you prefer to work in some more restrictive setting, that's perfectly fine. However, you can't then go on to proclaim that others who don't agree to also work in this more restrictive setting are somehow wrong – that's just silly. Within normal (ZF) set theory (and even in many other, less common), any equivalent construction of the real numbers is going to include as a true statement the fact that 0.999... = 1. The arguments are basic and easy to verify.
Again, your problem seems to be not with this, but with other foundational issues. You're then trying to say within your own framework that the statement "0.999... = 1" is false. Whether that's because of the actual framework, or simply because you're interpreting the statement differently than everyone else is irrelevant. You can't change the rules and then tell everyone else that they're wrong because they're not following your rules. If you want to show the falsity of the statement, you have to do it within the rules under which the claim is being made. Anything else is completely pointless. –Deacon Vorbis (carbon • videos) 17:29, 9 May 2019 (UTC)
I'm new to editing Wiki pages so thank you for pointing out my editing mistakes.
Formalism and Platonism are inseparable. Anyone who is a Formalist must also be a Platonist by necessity. Formalism is the viewpoint that 'mathematical knowledge' is gained through using rules to manipulate physical symbols. But any given collection of squiggles on a piece of paper has no inherent meaning. The formalists have to agree on what the different symbols mean. Some symbols might be called 'numerals' and others might be called 'operators' and so on. These meanings have to be conveyed using a natural language, and so the symbols are merely shorthand for some natural language meaning. Sadly natural language can include logical contradictions such as 'a married bachelor' or 'infinitely many' or 'we can physically work with things that are completely detached from physical reality'. But just because we CAN assign a contradictory meaning to a symbol, it doesn't mean that we should.
Formalists maintain that their mathematical objects and rules have nothing to do with the real world. This belief that mathematics is somehow detached from physical reality is Platonism. Therefore if someone claims to be a Formalist then by necessity they are also conceding to being a Platonist.
About your philosophy of mathematics you said "it's just really pointing out the distinction between the string "57" and the real number it represents, which is not a string of symbols, but rather an equivalence class of Cauchy sequences of rational numbers". We can easily say these words, but we cannot easily know what they mean. Nobody has any experience of anything that is 'not finite' and so I claim nobody really understands what one of these equivalence classes is.
We can experience endless algorithms such as: While 1=1: Print "Hello". We can also experience a large body of objects that fade out in the distance, and where we can't see an end point. We can also experience the counting of natural numbers. We know that if we are given the symbol for any natural number, then (if enough physical resources are available) we should be able to add 1 to it and construct the symbolic form of its successor. We might think that knowledge of all these concepts somehow enables us to understand what 'infinitely many' means but it doesn't. None of these things can be described as 'not finite'. Nobody has any concept of what 'not finite' means, but we still create definitions and rules and we pretend that this means we can work with the concept.
We don't even have a clear unambiguous agreed definition of exactly what mathematics is. I favour Bertrand Russell's description: "Mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true."
When we say something is mathematically proven, all this means is that a statement is valid according to a certain given set of rules and premises. But these rules and premises are allowed to be meaningless or even completely invalid; they can be any old nonsense. Mathematicians can therefore have great fun publishing loads of meaningless theorems and proofs.
You said "any equivalent construction of the real numbers is going to include as a true statement the fact that 0.999... = 1" but you have not responded my points on this subject.
You said "Again, your problem seems to be not with this, but with other foundational issues. You're then trying to say within your own framework that the statement "0.999... = 1" is false". I do have many issues with the foundations, but I believe 0.999... does not equal 1 within your framework, not one of my own.
To that end, let's discuss actual proofs starting with the 1st formal proof that appears on the main Wiki page. In my opening comment of this thread I claim to have exposed the flaw in that proof. If you can, please explain to me what is wrong with my argument. PenyKarma (talk) 01:08, 10 May 2019 (UTC)
There's no flaw in the proof. The x required in the proof is a fixed number; it doesn't depend on n. On the other hand, you're giving a whole sequence of numbers, and trying to sneak in a different value for x depending on n. This isn't what's being demanded in the proof, so it doesn't demonstrate anything.
For what it's worth, I don't think this is really the most instructive or clear proof here. The one based on the Cauchy sequence definition is much more straightforward. Proceeding like this also has the advantage that you don't need to consider any special properties of the real numbers to complete the proof. Instead, all the heavy lifting is done ahead of time when you first show that the Cauchy sequence construction describes a complete, ordered field, as we're looking for.
Once that's done, all you have to do is decide what's meant by "0.999...". There's more than one way to proceed here, but most people would agree that the most reasonable interpretation is the real number which is the equivalence class of Cauchy sequences represented by (0.9, 0.99, 0.999, ...). And similarly, "1" means the real number represented by the Cauchy sequence (1, 1, 1, ...). To show that "0.999... = 1" then means to show that the two representatives that we've chosen lie in the same equivalence class. This is done by showing that their termwise difference converges to 0. Indeed, that difference is the sequence (0.1, 0.01, 0.001, ...). This sequence does indeed converge to 0 (straightforward exercise for the reader), which means that the two sequences are in the same equivalence class, which means that "0.999..." and "1" represent the same real number, by definition. Short and sweet. –Deacon Vorbis (carbon • videos) 02:31, 10 May 2019 (UTC)
You said "There's no flaw in the proof. The x required in the proof is a fixed number; it doesn't depend on n.". So your objection to my argument appears to be that the x in the proof is not a real number (because many real numbers obviously CAN be described in terms of n, just like 0.999... can) but that x is a 'fixed number', whatever that is.
If I assume that by 'fixed number' you are referring to a fixed point data type, which is essentially an integer that is scaled by a certain factor, then the proof only applies to a subset of the real numbers. So all that it proves is that SOME real numbers cannot be placed between 0.999... and 1. My counter argument still holds that other real numbers CAN be placed between them.
The description on the main Wiki page is slippery in that it doesn't explicitly describe what type of number x is. You are claiming that it is not any real number but that it is any of a particular subset of the real numbers. As such, it only proves that numbers from that subset cannot be placed between 0.999... and 1.
Next you said "I don't think this is really the most instructive or clear proof here. The one based on the Cauchy sequence definition is much more straightforward.". By describing it as not the most instructive or clear it sounds like you don't put too much stock in its validity. You previously said "There's no flaw in the proof" and so I think it is important that we get to the bottom of this lack of clarity so that we can both agree on whether or not the proof is valid within your framework of mathematics. Can you confirm what number type you believe x to be? Is it ANY real number? Is it ANY fixed-point decimal (& therefore only a subset of the real numbers)? Or is it something else?
Moving on. let's consider your preferred argument, which is that the term-wise difference between the two sequences appears to approach zero, and therefore 0.999... and 1 are equal by definition. To the lay person, this is far from a clear and instructive proof. Indeed, it took over 200 years after the introduction of infinite decimals before any of the worlds greatest mathematicians devised this argument. And all that it demonstrates is that if we are inventive enough then we can construct a series of clever sounding definitions so that both 0.999... and 1 happen to fall into the same categorisation.
It causes confusion for the lay person because the meaning of terms like 'sum' and 'equals' have been redefined to mean something completely different from the intuitive trivial meanings that we first learn as children. Furthermore it all rests on the validity of the limit argument, which is not accepted by some well known mathematicians such as Professor Normal Wildberger, Dr. Doron Zeilberger and others. Indeed, even the Ancient Greeks had an argument that causes problems for the limit approach which I have explained several times in this thread.
And so your preferred argument is surrounded by controversy within your own ranks. It is confusing to the lay person and far from clear or intuitive. Even the Wiki page itself suggests that the intuitive explanation is "If one places 0.9, 0.99, 0.999, etc. on the number line, one sees immediately that all these points are to the left of 1, and that they get closer and closer to 1." and "there is no number that is less than 1/10n for all n". So I think we should focus on the formal proof of this intuitive explanation before we dive into the mire of equivalence classes of Cauchy sequencs and limits. PenyKarma (talk) 11:03, 10 May 2019 (UTC)

I'm not sure how much more I can say that hasn't already been said; your objection to the proof isn't valid because you're using a different value of x for each n. That's not what's required in the proof, so you haven't demonstrated anything by doing so. "Fixed number" does not mean "fixed point data type" (whatever that means exactly); it means that it's a single quantity within the scope of the proof, and its value doesn't depend on any other variables (most importantly n in this case). My issues with the proof are with its exposition, not its validity. It's essentially treating "0.999..." to be the least upper bound of the set {0.9, 0.99, 0.999, ...}. This is guaranteed to exist because the real numbers are complete, and this value is assigned to x. Then, it goes on to prove that 1 is this least upper bound, and hence what's meant by "0.999...". This is all perfectly valid, even if it's not clear from the write-up here.

Take a step back for a second and forget the math. How likely do you think it is that entire generations of brilliant mathematicians (far smarter than either you or I) have missed something so utterly basic that it was refuted in a couple of lines by a completely elementary argument? And these are people that are trained in the study of this subject, to analyze logical arguments critically. Or maybe is it just a teensy bit more likely that you don't quite understand something fully? Now, that's not a mathematical argument, but it's worth considering.

There is no controversy among mathematicians any more than there's controversy among Egyptologists that the pyramids weren't built by aliens. (On a side note, Wildberger is a kind of a crank (which I realize isn't an argument, but I really don't want to get into that here), and I suspect you're misrepresenting Zeilberger's views (who isn't a crank, but probably isn't saying what you think he's saying)). In any case, back to what I said earlier, even if there are mathematicians that (maybe due to philosophical views) prefer to work in some more restrictive settings which don't admit constructions of the real numbers, it makes absolutely no difference, because the original claim isn't being made in the more restrictive setting; it's being made within the usual foundations of mathematics.

And finally, your complaints about the Cauchy sequence argument being confusing are toothless (if maybe a bit telling). It is clear and instructive to someone with the background to digest it. Any formal proof is going to rely on a either a construction of (like via Cauchy sequences, or Dedekind cuts, or any number of others) or abstract characterization of (as a complete, ordered field) the real numbers. And any such approach is going to require a comparable level of mathematical sophistication that's not possessed by the layperson. It takes some work to get there, and you can't expect to learn it all in an afternoon. But with dedication, it can be learned. –Deacon Vorbis (carbon • videos) 13:39, 10 May 2019 (UTC)

To PenyKarma: I agree with everything that has been said by Deacon Vorbis. I would add some more general comments. It seems that you confuse the philosophical concept of truth with its mathematical counterpart. Platonism has to do with the philosophical concept, and has nothing to do with modern mathematics. A mathematical result is true only if it can be proved from the axioms of the theory in which is stated, and this has nothing to do with any physical interpretation. The mathematical notion of a proof is completely formalized, and there are software that allow verifying difficult proofs. On the other hand even the best computer scientists cannot imagine how verifying a philosophical truth on a computer. So involving philosophers about mathematical truth, as you did by referring to Platonism, is a fundamental error.
My second point is that there are deep philosophical questions about mathematics, about which there is no consensus, even among mathematicians. Unfortunately these questions are rarely discussed by philosophers. One of them is the following: Until the end of the 19th century the development of mathematics was mainly motivated by the study of the physical world. Since the beginning of the 20th century, many mathematical concepts and theories have been developed independently of any application, as there were motivated only by questions of pure mathematics. Nevertheless many such theories appeared later to be useful in physics. One famous example is the use of non-Euclidean geometry by Einstein, but many other examples are available. This set the question of what is the true relation between mathematics and the real (physical) world, and why pure mathematics are so useful. The answer of this important question can certainly not be found by classifying, as you did, thinkers into Platonists, modernists, formalists, post-modernists, etc. D.Lazard (talk) 15:33, 10 May 2019 (UTC)
First off, my arguments are not new, they have been around for over 2,000 years. Zeno devised some paradoxes that he claimed showed that time and/or movement could lead to contradiction. Democritus and some others noticed that these problems did not necessarily have to relate to motion or the passage of time, and this led to the foundation of Atomism. It was Democritus and some of his contemporaries that interpreted Zeno's paradoxes as showing that the concept of infinite divisibility leads to contradiction and therefore everything must consist of a finite amount of indivisible parts. These are exactly the same contradictions that come with the notion of pi (or any number) as being a constant on a continuous number line. It means that the concepts of real numbers and the continuum lead to contradiction.
For pi to exist on the number line, then the number line from zero to pi must be able to be considered as being a series of lines of lengths 3, 0.1, 0.04, and so on. Furthermore, the line from pi to the point 4 must also be able to be considered as a line on the continuous number line. And since this 'pi to 4' line must be connected to a line immediately before it, then there must be a 'last part' in the series of infinitely many lines from zero to pi. This forms a contradiction because the concept of 'infinitely many' parts requires there to be no last part.
It was in the 16th century when Simon Stevin created the basis for modern decimal notation in which he allowed an actual infinity of digits. Yes they knew about the contradictions of infinite division, but everyday mathematics used in businesses was made much easier by the widespread use of base 10 decimals. To my sceptical eye, it looks like mathematical rigour was sacrificed in favour of ease-of-use.
The original idea behind infinite decimals was that they were the sum of their rational parts. Essentially a real number was defined as being its decimal representation, the two were inseparable. This definition was considered inadequate by many, not least because its lack of uniqueness (as in 0.999... and 1 being the same number).
It was not until the early 19th century that limits and convergence were introduced. The equivalence class of Cauchy sequences finally gave us a unique construct for any one real number. Since it took over 200 years before any of the worlds greatest mathematicians devised this approach, it was clearly not intuitive at the time.
You said "Take a step back for a second and forget the math. How likely do you think it is that entire generations of brilliant mathematicians (far smarter than either you or I) have missed something so utterly basic that it was refuted in a couple of lines by a completely elementary argument? And these are people that are trained in the study of this subject, to analyze logical arguments critically. Or maybe is it just a teensy bit more likely that you don't quite understand something fully?". You could shorten this to "go away you stupid person".
You said "your complaints about the Cauchy sequence argument being confusing are toothless (if maybe a bit telling). It is clear and instructive to someone with the background to digest it.". I read this as you telling me that I find equivalence classes of Cauchy sequences troublesome because I'm not clever enough to get my head around it. You are right, I openly admit I cannot conceive of infinity.
For the real number 57, its equivalence class will contain the sequence whose nth term is 57 – 1/10n and the sequence whose nth term is 57 – 1/20n as well as infinitely many other sequences. Yes I struggle to get my head around conceiving infinitely many of something, especially when I am aware of the contradictions associated with 'infinitely many' highlighted by the Atomists.
I am fully aware that on the cosmic scale of cleverness I am a mere infinitesimal distance from the bottom. I'm sure you have already explained to me as simply and clearly as you can why the Atomist argument is flawed, but I persist because I am just too stupid to understand it. I only studied maths up to A-level and then a little more at University whilst studying Computing Science. I guess this is not a good enough maths background to understand why 0.999... equals 1.
It is testament to my stupidity that I apparently don't even understand the elementary proof. The Wiki page introduces the proof thus: "There is an elementary proof of the equation 0.999... = 1, which uses just the mathematical tools of comparison and addition of (finite) decimal numbers, without any reference to more advanced topics such as series, limits, formal construction of real numbers, etc. ". I would have expected the formal version to also avoid reference to more advanced topics but you have just told me about it "treating "0.999..." to be the least upper bound of the set {0.9, 0.99, 0.999, ...}" and "This is guaranteed to exist because the real numbers are complete". If the proof already accepts the definitions of real numbers and the completeness of them, then what is left to be proved?
When I try to understand the proof without reference to advanced concepts, it appears to me to be a statement about infinite decimal representation. It is all about what you can fit into n decimal places using a decimal system. If n is 5 then we cannot construct any decimal with 5 decimal places that is between 1.00000 and 0.99999, and this holds for any value of n. This is all it says to me.
I read it as a proof by contradiction where we start by assuming that a unique number is defined as its infinite decimal representation with no leading zeros in front of the units column. This means that since 0.999... and 1.000... have different decimal representations, we assume that they are different numbers by definition. We also assume that an infinite decimal representation is a coherent concept that does not lead to contradiction. We also assume that any fractions (e.g. 1/3) can be fully represented by an infinite decimal.
Then the proof leads to contradiction and so this means that one or more of our assumptions must be wrong. The popular interpretation of the contradiction is that it simply means that our assumption that decimal representation is unique must be wrong. But if we adopt this solution then we will be ignoring the fact that our assumption that an infinite decimal is a coherent concept still has the unresolved issues highlighted by the Atomists. PenyKarma (talk) 18:39, 10 May 2019 (UTC)
To Deacon Vorbis: Please note that I did not try to change your signature, perhaps our edits clashed or maybe I placed my signature in the wrong place? I am trying to revise something that I said 15 days ago, but I am not trying to change the meaning. I'm just trying to add clarification for any 1st time readers. It would have exactly the same meaning but it would read better for new vsitors. It is important that it is easy to understand because it relates to my objection to the proof. Any mathematician would realise that the change is not substantive because there is no change of meaning in terms of the mathematical argument. Would you be happy with this change?... Just before I say:
We can use the nth sum of x = 1 – 0.5/10n
I'd like to add this:
For example. let x = 95/100 + 45/1000 + 45/10000 + 45/100000 + … PenyKarma (talk) 18:08, 24 May 2019 (UTC)
If you want to add to something you said 15 days ago, after it's been more than responded to, then add it at the bottom. Hopefully, any first time readers aren't misled by the nonsense that you continue to spout. I've been more than patient explaining where you're mistaken, but you refuse to listen, having already convinced yourself of your own inerrancy. I'm done here. –Deacon Vorbis (carbon • videos) 18:14, 24 May 2019 (UTC)
You said "your objection to the proof isn't valid because you're using a different value of x for each n."
But it is obviously a different value because it is the nth partial sum of 95/100 + 45/1000 + 45/10000 + 45/100000 + … just like the proof uses different values (i.e. partial sums) of 9/10 + 9/100 + 9/1000 + ...
I took your comment on board and I said that the only other way I could interpret the proof is if it only relates to decimal representations where n is the nth decimal place. In that case my objection is that we cannot assume that all rationals (or sums of rationals) can be represented by a decimal representation. That would be to assume things like 1/3 equals 0.333... and this is precisely equivalent to what we need to prove.
Now you have resorted to insults and you ended with "I'm done here". Given the tone of your last comment I'm glad your done. You think my lack of intellect is justification for you to insult me. You are wrong. There is no excuse for your behaviour. PenyKarma (talk) 18:51, 24 May 2019 (UTC)
OK, there's nothing going on here. This article is about 0.999... in the real numbers; the real numbers contain no infinitesimal, which leads inexorably to the conclusion that 0.999... is equal to 1. If you wish to work in some other philosophical system, feel free, but not here. --jpgordon𝄢𝄆 𝄐𝄇 20:35, 24 May 2019 (UTC)

For pi to exist on the number line, then the number line from zero to pi must be able to be considered as being a series of lines of lengths 3, 0.1, 0.04, and so on. Furthermore, the line from pi to the point 4 must also be able to be considered as a line on the continuous number line. And since this 'pi to 4' line must be connected to a line immediately before it, then there must be a 'last part' in the series of infinitely many lines from zero to pi. Nope. @PenyKarma: You’re never going to be taken seriously here since literally none of what you said there is well-defined, making your entire argument just hand-waving.—Jasper Deng (talk) 21:50, 25 May 2019 (UTC)

You see Jasper Deng, this is why this argument never ends. What you have basically said here is "I don't understand your argument, therefore you are wrong." It's perfectly clear to me what PenyKarma is saying, so I can't imagine why you would think your reply is persuasive. Algr (talk) 14:51, 30 January 2020 (UTC)
@Algr: Strawman argument. We cannot even evaluate the truth of statements that are not even wrong; notice how I made no explicit pronouncement on the truth of his statement.--Jasper Deng (talk) 09:19, 19 February 2020 (UTC)
Jasper Deng At least a strawman argument claims to try to understand what was being said. You haven't even done that. "Not even wrong" is more appropriate to your statement because you don't actually say anything about .999... You are just engaging in fancy name calling. Algr (talk) 19:27, 20 February 2020 (UTC)
Nope. "Not even wrong" applies wholeheartedly to PenyKarma's argument since they are devoid of rigorous meaning. Specifically, "And since this 'pi to 4' line must be connected to a line immediately before it, then there must be a 'last part' in the series of infinitely many lines from zero to pi." is meaningless; in fact, the second half of it is self-contradictory in any reasonable interpretation. What does he mean by "line"s? There's nothing about the real line that asks for this. @Algr: Considering that you have for many years demonstrated that your understanding of this subject is woefully inadequate to converse here, please stay out of any further conversations here.--Jasper Deng (talk) 21:28, 12 March 2020 (UTC)
My line argument is easy to understand. The complaint about what I mean by 'line' is just nitpicking because I did not use the more precise expression 'closed line segment' (which includes both end points).
The closed line segment from 0 to 3 shares just one point with the closed line segment from 3 to 3.1. Apart from the overlapping point, these two line segments equate to the single closed line segment from 0 to 3.1. If the decimal value for pi can exist on the number line, then it follows that each of the line segments that I described earlier (0 to 3, 3 to 3.1, 3.1 to 3.14 and so on, forming infinitely many line segments) must also be able to exist as their static start and end points must exist on the number line.
It then follows that the line segment from pi to 4 must share the point 'pi' with just one of the infinitely many line segments described earlier. In other words, it must connect to a last line segment within the infinitely many line segments. This forms a contradiction as 'infinitely many' requires there to not be a last line segment. The same argument could be made with 0.999... instead of pi. The concept of an infinite decimal always leads to contradiction.
Those who have an unshakable belief in the mystical concept of mathematical infinity will always construct slippery, murky, and over complicated arguments in a futile attempt to justify it. These arguments include all the so-called proofs for 0.999... equals 1. PenyKarma (talk) 14:07, 14 March 2020 (UTC)
@PenyKarma: Here's your fallacy then: you have discovered that the union of infinitely many (even countably many) closed sets is not necessarily closed and there are no reasons to believe otherwise. Their union is a half-open interval including 0 but not pi, since by definition of a set union, pi would have to belong to at least one of the sets in question, but it does not. There's nothing paradoxical about that and it does not disprove the idea of "infinity". Any closed interval from pi to some greater number will have empty intersection with this half-open interval and yet no number in the union of this interval with all those intervals will be omitted (so in your example, the union of all these is still the closed interval from 0 to 4). But the least upper bound of the union of all the intervals you mentioned that are less than pi is still pi, and that is the definition of a decimal representation. Sorry, but you're wrong again!--Jasper Deng (talk) 20:43, 14 March 2020 (UTC)
@Jasper Deng: In my example, all of my lines are closed line segments with a well defined point at each end. They are ordered and, going from left to right (in relation to their mapping on the number line), the end point of one line is also the start point of the next line.
But with your half-open interval argument you appear to be claiming that none of the infinitely many closed line segments (from 0 to pi) in my argument can contain the point pi. In other words, you are saying that if the infinite decimal corresponding to pi could exist, then the sum of all its digits would not reach pi. You are effectively saying that pi does not equal pi. This supports my claim that infinite decimals cannot exist.
My line argument makes sense to many non mathematicians. They can see that there is an obvious contradiction. The counter arguments presented by mathematicians are always something like your least upper bound interpretation of a decimal representation. They are nothing more than slippery wordplay. If you could actually determine the least upper bound (which you can't in this case because of its infinite nature) then you are back where you started with an infinite decimal. And so you can't actually describe the infinite decimal for pi this way as it is a circular argument at best. The messy and complicated counter arguments might sound clever but they resolve nothing. The contradiction is still there. It is clear and obvious, unlike the counter arguments. PenyKarma (talk) 00:35, 15 March 2020 (UTC)
@PenyKarma: Sorry, you completely ignored the part about the least upper bound. If you want more detail on that, see the Dedekind cut construction of the real numbers. In particular, if one bounded set is the closure of another, then their least upper bounds are equal. In particular, . That your view is absurd is demonstrated by the effect of changing base to base 2, 3, etc. "Pi" is not equal to the value of any finite truncation of its decimal expansion but is the supremum of the set of all such expansions. You also clearly have no clue what you are talking about when you say "reach"; "reach" here means "converges to" and for an increasing monotonic sequence like this one that means taking the supremum which need not be a member of the sequence itself. This is how decimal expansions work and therefore, your line argument is nothing but complete bullocks in the real numbers. There is absolutely nothing whatsoever that requires the union of all these closed line segments to be closed. We are not bound by physical limitations on however many "lines" there are. So please, stop wasting your own time on this useless argument and learn some actual real analysis. I stand by my earlier dismissal of your argument even more after this nonsense).--Jasper Deng (talk) 01:28, 15 March 2020 (UTC)
@PenyKarma: And "least upper bound" is not "slipipery word play". You are so blinded by your refusal to actually learn real analysis it's not even funny. Completeness (which the least upper bound is one form of) is one of the most fundamental properties of the real numbers. If you are going to reject that, then you cannot possibly be talking about the real numbers. In that case, please do us a favor and leave, because there is nothing more to be discussed.--Jasper Deng (talk) 01:36, 15 March 2020 (UTC)

The discussion above is closed. Please do not modify it. Subsequent comments should be made on the appropriate discussion page. No further edits should be made to this discussion.