Talk:Berry paradox

2003

However, the less formal version given above can apparently be solved by simply noting that there is nothing to stop us from giving any name we please to any number. For example, the enormous number of 10 to the power of 100 is called a googol. So, there is no smallest positive integer not nameable in under eleven words.

The above paragraph is rather missing the point. Although not saying anything untrue, it doesn't really give a resolution to the paradox. A name phrase like "the smallest positive integer not nameable in under eighteen words using words current during the nineteenth century" is also a valid example of the Berry paradox but it can't be resolved by using the "naming after the fact" method described above. -- Derek Ross 03:41 May 9, 2003 (UTC)

Hmm. But I can still name any number after a word current in the nineteenth century. There's nothing to stop us calling any number anything we please. So we can call it "Joe", or whatever. Evercat 03:45 May 9, 2003 (UTC)

Of course you can, but once you have assigned every 17 element combination of any name and any word ever spoken in any nineteenth century language including all the nonsense words that nineteenth century children invented, you will still be left with unnamed numbers, one of which will fit the paradoxical definition above. That's because the number of nineteenth century words is finite whereas the number of positive integers is infinite. And since you will have used up all the 17 element combinations of nineteenth century words, you will not be able to give the number a name consisting of less than 18 words. -- Derek Ross 03:53 May 9, 2003 (UTC)

Do I need to name every number in existence? Isn't it enough just to potentially be able to name any number that might otherwise be the Berry number? I mean, the point is supposed to be that there is no Berry number, because any number can be named with just one word. Anyway, if you're confident you're correct, feel free to revert or edit in the objection. Evercat 21:17 May 9, 2003 (UTC)

Unfortunately to use the given method to resolve the Berry paradox, you do need to be able to name every number in existence. Look at how it works. First I identify the Berry number by some slow but sure means -- let's say simple search through all the named numbers. Eventually I'll find a number which can't be named in less than 18 words (except by using the Berry phrase). Second I give it a name. Have I resolved the Berry paradox ? No, because if I repeat the process I now find that there is still a number which can't be named in less than 18 words (except by using the Berry phrase). It's a bigger number than the one I just named, of course, but one can still be found. So I give it a name. Have I resolved the Berry paradox ? Well, no. It doesn't matter how many times I name the current Berry number. There's always another bigger potential Berry number, waiting out there to replace it. To get them all, I need to be able to name every number in existence. You could say that there are an infinite number of words that can be invented and that's true but if I change the Berry sentence to something like the nineteenth century example above or, even more simply, to a form like "The smallest positive integer not nameable using less than eighty-nine ASCII characters", the naming trick won't work. -- Derek Ross 23:04 May 11, 2003 (UTC)

What's to stop us using the same name more than once? As long as we don't do so at the same time. Evercat 23:11 May 11, 2003 (UTC)

The point is that you have to name the numbers so that they are uniquely recognisable, i.e. if you name them and then explain your system to me then I should be able to tell which number a stream of words represents. Thus, naming both the numbers 10 and 12 Bob fails this criteria as if the word Bob is written down I have no way of knowing if the number is 10 or 12. -- Ams80

Anyway, here's a thought. What's the significant difference between the Berry Paradox and something like: "The atom not nameable in under 9 words" or whatever? There is no such atom, despite the huge number of them. They can all be given names. Just not at the same time. But so what? Nameability just requires the potential for that one object to be named, and every other object is irrelevant. Isn't it?

The above was written before Ams' comment. My answer to that is the same though, nameability is all about the possibility of that one thing being named. Everything is nameable. Evercat 23:20 May 11, 2003 (UTC)

Unfortunately not. Only the imaginable can be named. If you can't imagine it, you can't name it -- (and yet it may well exist). -- Derek Ross 00:27 May 12, 2003 (UTC)

---

Please note that the Berry Paradox emerges "naturally". Consider the very old proof that there are an infinite number of primes, and notice also that in the life of the human species only a finite number will be/have been found; call these prime' numbers. Then in an attempt to resolve this contradiction go over the traditional proof substituting prime' for prime throughout. Then note the false statement(s) in the revised proof. No doubt there are other places the Berry Paradox could emerge. PML.

---

However, the less formal version given above can apparently be solved by simply noting that there is nothing to stop us from giving any name we please to any number. For example, the enormous number of 10 to the power of 100 is called a googol. So, there is no smallest positive integer not nameable in under eleven words.

How is this a disproof, provided we have a language with words limited to some finite length?

Because you don't need to actually name every number - all that matters is that every number is nameable, ie can be named. Everything can be named. Not all at once, but that's irrelevant. Evercat 12:53 30 Jun 2003 (UTC)

My intuition of "named" does not extend to this. I think in terms of what might be written as "putting names into a one-to-one (or many-to-one) correspondence with named objects". Given any single naming scheme, the above does not hold water. If you can switch naming schemes on the fly, practically anything can be proved or disproved... Karada 13:08 30 Jun 2003 (UTC)

Example "one = two". On the left hand side, "one" denotes 1, and "two" denotes 2. On the right hand side, "two" denotes 1, and "one" denotes 2. "But that's silly", I hear you say, "you can't just change notation in the middle!" Exactly. -- Karada 13:21 30 Jun 2003 (UTC)

2004

If today (14th sept. 2004) you call 10^999999999999999 "Bob", then "Bob" means 10^999999999999999 ONLY in 2004 English and (maybe) later ones, but it doesn't mean 10^999999999999999 in 19th Century English. Unless you are sure that somebody called a number "Bob" in the 19th century, you can't say that "Bob" is a 19th Century English word for any number.Army1987 17:21, 14 Sep 2004 (UTC)

2005

MathWorld Entry

At Wolfram Research's MathWorld.com website the Berry Paradox can be found at this location: http://mathworld.wolfram.com/BerryParadox.html. It seems to not have the restriction to positive integers. Is this restriction necessary to maintain on Wikipedia?

That definition uses the phrase "least integer" - I'd contend that "least" implies quantity, which in turn implies non-negative numbers. If it had said "lowest integer" on the other hand, that would be a different matter, as there are an infinite number of very large negative numbers we can't possibly name within eleven words. I'm not sure whether this in itself invalidates the statement, but at minimum it adds an unnecessary complication to it. Slovakia 03:24, 4 December 2005 (UTC)

Another Different Naming Structure

And what about the fact that there are different ways of expressing any numnber? For instance, the number "one thousand five hundred and twenty one" could be easily be expressed as "five hundred and seven times three" - which is only 6 words, to the original 7.

Reffering to a number in this way doesn't require actually renaming the number, because this way of expressing it already existed. 65.95.245.90 06:51, 26 May 2005 (UTC)

note - the above is actually me, i just forgot to log in at the time! Oracleoftruth

Reading something like that in words, I'd be likely to understand 500+7×3=521, instead of 507×3=1521.--Army1987 22:00, 9 August 2005 (UTC)

It doesn't help you to avoid the paradox. In fact, it just serves to make you run out of names (or descriptions, or whatever you want to call them) faster. There can only be N^10 possible names, where N is the number of words in the English language, so Berry's number is some figure below N^10. Slovakia 03:24, 4 December 2005 (UTC)

2006

21, not 91

The smallest positive integer not nameable in under two words is 21, not 91, right? --ChadThomson 09:20, 30 November 2005 (UTC)

• Just came on here to ask that myself. The phrase "A reasonable definition of English" apparantly reshapes the berry paradox into "The smallest positive integer not nameable in under two words, whose subsequent integers also possess this quality," which is stretching it a bit :P GeeJo _{(t) (c)} 11:58, 2 December 2005 (UTC)

Ditto. I've decided to Be Bold and fix it (and in the process drop in an extra hint about why the two-word case is not problematic, because the statement doesn't satisfy its own criteria). Slovakia 03:24, 4 December 2005 (UTC)

The basic idea of the proof is that a proposition that holds of x if x = n for some natural number n can be called a "name" for x.

I'm taking the liberty to change this to a name for n . --62.219.170.138 08:21, 5 April 2006 (UTC)

Continuing the 2004 discussion

I appreciate that this debate seemingly ended ages ago, but I thought I would contribute my answer to this merely for the sake of anyone, like me, who decides to take a gander at the discussion, particularly those similarly confused with this paradox as it appears many people are.

The simple fact is that "naming numbers with random words", ie "Joe" etc as previously used for this example, is entirely beside the point. The point is that the statement we all try to disprove - in the original paradox, it is "The least integer not nameable in fewer than nineteen syllables" has been specifically designed to be a phrase which goes under the required syllable (or word, etc) count. It doesn't matter at all about any number which can be named. Consider any number which you know and can easily proove that you name in under nineteen syllables to be discounted. Now take the lowest number out of what remains. That number then becomes defined as "the least integer not nameable in fewer than nineteen syllables". But in becoming that number, it has become a number which can be defined in under nineteen syllables, so it is discounted. Thus we move onto the next-lowest. But doing this means that number can now be defined in less than nineteen syllables, so we discount it and move onto the next. This process forces us to discount every single number we come across, ad infinitum - at the end, after an infinite number of discounted numbers, we find that no number has not been discounted, and so there is no number which cannot be defined in under nineteen syllables.

Again, sorry to anyone who takes offense to or finds annoyance in my contributing to a discussion which ended so long ago, but I believed that this discussion was in need of another explanation. That, and I do enjoy the challenge of explaining something like this which others struggle on. I would edit the actual article to make it more clear, but I'm not sure I could do the article justice. Anyone else who reads this and feels up to the challenge is welcome to use anything I just wrote as part of their edit, though, if it helps. Falastur 03:16, 14 April 2006 (UTC)

2007

In the above discussion it was said that "The point is that you have to name the numbers so that they are uniquely recognisable." In the Journal of Symbolic Logic, Vol. 53, No. 4, 1220-1223. Dec., 1988, in "The False Assumption Underlying Berry's Paradox," James D. French demonstrated that an infinite number of numbers could be uniquely described in the exact same words. French, 04 January 2007

Alternative explanation

Added in an alternative explanation of Berry's paradox. I feel the explanation using "set A" could be a bit confusing for people who do not have much knowledge of mathematics.

Saurabhb 23:09, 26 February 2007 (UTC)

Unfortunately, it's not that simple. Your "alternative explanation" explains why nothing can match the description "The smallest positive integer not definable in under eleven words", but that's hardly a paradox; there are many hypothetical descriptions that nothing matches, such as "The smallest positive integer less than zero", and most cause no consternation. What makes the Berry paradox a paradox is that something must match that description, yet nothing can. I really don't think there's a way to explain the full paradox without making reference to naive set theory, but you're welcome to give it a shot. —Ruakh_TALK 04:57, 27 February 2007 (UTC)

The assumption that "something must match that description" is a false assumption as demonstated by the paradox itself. WAS 4.250 16:33, 11 May 2007 (UTC)

Missing the point

(Sorry if someone has already explained this but) I think that some of you have missed the point. you do not need to go as far as 10^999999999999999 (as in one of the above examples). The paradox explicitly states the use of 'words' when searching for the number. Here is a number "One Hundred and twenty one thousand one hundred and twenty one" this is a number that is not definable in under 11 words (it requires 11 words). I am sure that there will be a smaller number that also requires 11 words or more, find it!

No I won't find what you ask me to find, instead I'll contradict your previous statement: "One Hundred and twenty one thousand one hundred and twenty one" is no more, no less than "eleven times eleven times one thousand plus eleven times eleven". Which is 10 words. Hence the number can be defined in less than 11 words, and I'm too good. ThorinMuglindir (talk) 13:05, 11 May 2013 (UTC)

Once you have found that number, you (in theory) have found The smallest positive integer not definable in under eleven words.

The paradox occurs by the fact that though the number could not be defined in under 11 words, it can be defined in under 11 words by refering to the number as "The smallest positive integer not definable in under eleven words" (which defines it in under 11 words, because the phrase is only 10 words long)!

—The preceding unsigned comment was added by Cs1kh (talk • contribs) 12:28, 19 July 2007 (UTC).

Semantics

"It is generally accepted that the Berry paradox results from interpreting sets of possibly self-referential expressions: it and similar paradoxes embody so-called "vicious-circle" fallacies. To resolve one of these paradoxes means to pinpoint exactly where our use of language went wrong and to provide restrictions on the use of language which may avoid them"

It is my belief that the entire mess lies simply in a weak definition or even misuse of the word "word". I do not see why a language must or should contain a finite number of words, only a finite number that can be expressed. It is, however, so with primes, though that is no reason to discount their multitude. While the above discussion makes the point that one might call a number "Joe" and in so doing define it, such nomenclature is really quite unnecessary, for numbers in themselves are words. Eight billion four thousand sixty two and nine millionths may very easily be considered a single word despite the spaces, just as The United States of America is oft considered a single word. With this is mind the only real constraints on naming numbers are their ability to be expressed and their system of classification. The paradox did say "can be" expressed, and so the first constraint isn't a problem at all as it would be assumed that a finite word could be expressed given enough time. The second constraint is classification. We must construct a system that both does not break down with degrees of magnitude and can express any number finitely that the previous constraint be non-constraining. (talk ) —The preceding signed but undated comment was added at 20:05, August 23, 2007 (UTC).

Just a note: the example given in the wiki-page ("The smallest positive integer"...) explicitly states/requests 'integer'. can you re-explain using only integer(s) instead of subsections/fractions (e.g. millionths) ([[User:Cs1kh]] (talk) 13:18, 7 February 2008 (UTC))

Explanation of the paradox

The paragraph explaining the paradox is rather vague; by the time that it points out that the defining phrase is itself less than eleven words long, the reader is already six lines into the explanation. This needs rewording for clarity. -- Sasuke Sarutobi (talk) 17:39, 23 May 2008 (UTC)

Article's Self Critique

The article currently has a self-critique that contradicts another part of the article:

The argument that "Since there are infinitely many positive integers, this means that there are positive integers that cannot be defined by phrases of under eleven words" assumes that "there must be an integer defined by this expression" which is counterfactual as most phrases "under eleven words" are ambiguous to their defining of an integer, with this ten word paradox being an example. Assuming one can match word phrases to numbers is a mistaken assumption.

As an article should be self consistent, we should address the critique and rewrite the article. (aside: Should an article not critique itself? Note boilerplates are about articles and not a part of them, and thus can critique articles.) I have a few questions and points to start discussion:

• Where the critique says "there must be an integer defined by this expression", does "this expression" refer to the statement of the Barry paradox here called B: "The smallest positive integer not definable in under eleven words"? Does it refer to the property ~D(n): "n is an integer not definable in under eleven words"? Does it refer to D(n): "n is an integer definable in under eleven words"? Some other expression?
• Why does the argument "Since there are infinitely many positive integers" assume there is an integer defined by the expression? Isn't it deducing it from the preceding statement which uses the pigeonhole principle: "Since there are finitely many words [...]"? Perhaps this first statement is making an assumption that an N exists such that D(n).

French's 2007 comment about an infinite number of uniquely described numbers applies somewhere to the critique. I'd like to hear more about it (as I haven't yet been able to obtain a copy of the original paper). More thoughts to follow as I try to wrap my head around the critique.Kanenas (talk) 18:59, 27 July 2008 (UTC)

Resolution

"The number not nameable0 in less than eleven syllables" is 16 syllables long even if the "0" is silent, or more depending on how it's pronounced, so this is completely irrelevant to the paradox.

Also, the section is highly redundant. In particular, the 4th paragraph is almost a straightforward restatement of the 1st.

I'll try to fix it, but no promises. --76.202.59.43 (talk) 04:13, 25 February 2009 (UTC)

Obvious Solution

There is an obvious solution. It's called restate the statement in a way so the statement has more than 11 words. 24.1.201.172 (talk) 23:30, 6 June 2010 (UTC)

That is missing the point entirely. The paradox is that it is possible to state the in under 11 words, and even if it were stated in a more verbose way, it would still be possible to state it as it was originally stated.109.148.239.97 (talk) 17:35, 29 December 2012 (UTC)

Nonsense

In the "Resolution" section, there is the following non-sensical passage:

The argument presented above that "Since there are infinitely many positive integers, this means that there are positive integers that cannot be defined by phrases of under eleven words" assumes that "there must be an integer defined by this expression". This is counterfactual, as most phrases "under eleven words" are ambiguous to their defining of an integer, with this ten word paradox being an example. Assuming one can match word phrases to numbers is a mistaken assumption.

Certainly the first expression quoted makes no assumption whatsoever regarding definability of integers by eleven-word phrases. It merely makes the observation that, of the eleven-word phrases that do define some integer uniquely, there are only finitely many. That is certainly true, since there are finitely many phrases using eleven words of the English language (of which the unambiguous -- uniquely defining -- phrases form a subset). The corollary is that, indeed, there are only finitely many numbers uniquely defined by eleven-word phrases.

In light of this, I will hereby delete the offending passage.Computationalist (talk) 18:44, 10 August 2010 (UTC)

The "Resolution" is not one

The section titled "Resolution" does anything but resolve the paradox and should be deleted until someone comes up with an adequate explanation of the paradox. Its first paragraph reads:

"The Berry paradox as formulated above arises because of systematic ambiguity in the word definable. In other formulations of the Berry paradox, such as one that instead reads: "...not nameable in less..." the term "nameable" is also one that has this systematic ambiguity. Terms of this kind give rise to vicious circle fallacies. Other terms with this type of ambiguity are: satisfiable, true, false, function, property, class, relation, cardinal, and ordinal. To resolve one of these paradoxes means to pinpoint exactly where our use of language went wrong and to provide restrictions on the use of language which may avoid them."

How about elaborating on just how the "systematic ambiguity in the word definable" leads to the paradox -- instead of suddenly changing the subject to "other formulations of the Berry paradox" ??? First deal with *this* formulation, and not by a breezy dismissal like "systematic ambiguity" that avoids any attempt to actually communicate exactly how the putative reasoning in this paradox goes off track.

Then, once that is done successfully, by all means mention other formulations and generalizations to your heart's content. But please first address the subject of the article rather than anything but.Daqu (talk) 07:35, 6 February 2011 (UTC)

Not a paradox

Who says it's a paradox? It is just a proof that there is no smallest positive integer definable over 10 words. — Preceding unsigned comment added by 89.164.168.87 (talk) 01:44, 2 January 2012 (UTC)

There's a finite number of sentences formed from 10 words; if we say there's a million words, then that's at most 10^60 sentences and thus integers. Thus, by the pigeonhole principle, at least one of the first 10^60+1 integers is not describable in ten words. One of them has got to be smallest. Therein lies the paradox.--Prosfilaes (talk) 21:39, 16 August 2012 (UTC)

It's a colloquial symbol of a concept

Come now, isn't it recognized that Berry's Paradox in its original form is, in and of itself, too colloquial to be worth treating as a real information theoretical dilemma? It's the idea behind it which counts. This is why we have Kolmogorov Complexity. The core issue is of information content. The total information content in the original Berry's Paradox is part explicit (the sentence as written) and part implicit. The person considering the paradox will, herself or himself, implicitly contribute information.

By comparison, the total information content in a Turing machine isn't just the bits written on the tape; it's also the program inside the machine which tells it how to utilize the bits on the tape!

What's important about Berry's Paradox is that it represents something which actually does occur in concrete problems involving logic, information and computation. See Chaitin Incompleteness. If you want to think concretely and clearly about logic and information, use Turing machines and computation and bits, Goedel numbering, etc. One must quantify the total information in a fairly rigorous and standardized way so that nothing is left implicit. So, TL;DR: This kind of paradox does arise in a concrete sense involving Kolmogorov complexity and computation / logic.

If we want to make Berry's Paradox reasonable and rigorous, we must write down the Turing machine version of the problem, which we can objectively test and think about. First we need to consider that Berry's Statement is mal-formed if we want to put it into a computational format. It should read The least positive integer not generatable with input & program length totaling less than xxx bytes, etc.

152.3.68.83 (talk) 17:08, 16 August 2012 (UTC)

Assumes a one-to-one mapping

Maybe I'm missing something, but I could not find if it was stated that there has to be a one-to-one mapping from the combination of words to the integers, which is clearly what the section The paradox assumes. Since, if you first come up with the set you can normally define in less than eleven words, and then "all numbers previously not described in less than eleven words" and you've captured all of the numbers that were considered to be outside the set. If all it needs is the distinction between homomorphism and isomorphism, it's not really a paradox, you just have to set up the language correctly (not the same thing as what the Resolution section is talking about). Thus, it seems more like an ill-posed statement.

If I'm missing something important please let me know, I do not claim to be an expert on this; just curious. (Jahansen (talk) 05:41, 30 January 2014 (UTC))

You're not defining a set, you're defining an integer. Consider the text "a multiple of two which is less than pi" - this defines a single positive integer, two. But "a multiple of two which is less than two pi" gives the set {2,4,6}, which is not defining an integer. To define an integer you need to exclude all other integers. -mattbuck (Talk) 09:14, 30 January 2014 (UTC)

Rejection of argument by contradiction

I removed the following comment by an anonymous editor:

NOTE: Out of a contradiction one can prove anything. So, if you accept that "the expression is self-contradictory", you can conclude that "there can (or cannot) be any integer defined by it". What the original writer wrote, does not follow.

I'm posting it here in case anyone wants to discuss the topic. --Slashme (talk) 09:39, 29 February 2016 (UTC)

There is a proof against your argument. --5.249.127.17 (talk) 15:23, 28 July 2016 (UTC)

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Which paradox did Berry actually suggest?

The lede paragraph says:

Bertrand Russell, the first to discuss the paradox in print, attributed it to G. G. Berry (1867–1928), a junior librarian at Oxford's Bodleian library, who had suggested the more limited paradox arising from the expression "the first undefinable ordinal".

The second part of this sentence is uncited, and I believe it is false.

In the 1908 paper Mathematical Logic as based on the theory of types, which I believe is the original printed source for the Berry paradox, Russell discusses seven related paradoxes, including what have become known as the Epimenides, Berry, Richard, and Burali-Forti paradoxes. The one he attributes to Berry is paradox number 4, of which he says, in part:

4. … Hence, “the least integer not nameable in fewer than nineteen syllables” must denote a definite integer…

(You can presumably imagine the rest.)

The sentence I quoted from our Wikipedia article, however, seems to be referring to paradox number 5:

5. Among transfinite ordinals some can be defined… the total number of possible definitions is $\aleph _{0}$… there must be undefinable ordinals, and among these there must be a least.

(Again you can imagine how it proceeds.)

Russell does not attribute paradox #5 to Berry. Instead, he provides references to discussions in papers of König, Dixon, and Hobson from 1905–1906.

So contrary to the claim of our article, it appears that, at least according to Russell, the Berry paradox is not “the more limited paradox arising from the expression ‘the first undefinable ordinal’”. That more limited paradox (#5) had already been considered by several mathematicians. What Russell specifically attributes to Berry, paradox #4, is what is now known as the Berry paradox itself.

If there is no substantive objection, I will correct the lede paragaph shortly. —Mark Dominus (talk) 21:06, 2 August 2017 (UTC)

With less letters / lesslettered

smallestpositivinteger hyposixtiletterly non-definable — Preceding unsigned comment added by 2A02:587:4116:B300:80C3:2CCC:AD87:B7C1 (talk) 05:36, 20 April 2018 (UTC)

better form: smallestpositivinteger subsixtiletterly non-definable — Preceding unsigned comment added by 2A02:587:4116:B300:80C3:2CCC:AD87:B7C1 (talk) 05:39, 20 April 2018 (UTC)

Is English a dead language? Berry was wrong, English is entropic and alive! (the entropic principle)

Berry pseudoparadox

"The smallest positive integer not definable in under sixty letters" = the Berrinteger — Preceding unsigned comment added by 2A02:587:4116:B300:80C3:2CCC:AD87:B7C1 (talk) 06:01, 20 April 2018 (UTC)

1,121,373.

"One million one hundred twenty one thousand three hundred seventy three." 61 letters

I figured this out by asking Quora: https://www.quora.com/What-actually-is-the-smallest-integer-that-cannot-be-defined-with-English-numerical-words-in-under-sixty-letters

It might be wrong, but it seems like a decent enough answer.

;D

LionKimbro (talk)