Talk:Graham's number/Archive 3

Archive 1

Archive 2

Archive 3

Information in Graham's Number

The article includes the interesting comment that "Graham's number is "unimaginably" large in the sense that its digits contain more information than can be contained within the volume of a human brain..." If we're talking about information, in the sense of Kolmogorov complexity, then it doesn't really contain that much information at all. After all, every digit has been precisely and unambiguously specified in a medium-sized Wikipedia article. If we mean the digits themselves, we've already established that the digits of Graham's number cannot be stuffed into the known universe, so what's the point of mentioning that they won't fit into a human brain? Am I missing something? 159.172.43.6 (talk) 00:18, 23 February 2017 (UTC)

The note is only there because some ignorant people without the imagination to imagine that this concept is impossible for the human brain to imagine keep demanding a citation for the claim. That note was only put in to shut them up. The real meaning is that it is impossible to construct any analogy to illustrate the size of the number. Analogies like grains of sand in the desert, or stars in the universe are all completely inadequate. Any possible analogy contained within the universe will always be inadequate. SpinningSpark 00:30, 23 February 2017 (UTC)

I think that the IP is right, and this response is non-responsive. I also think the earlier criticisms of this word and note were correct, as well. The most recent removal was reverted on the ground that this has been "argued out at great length" here, but what is visible on the talk page is a two-editor consensus (that happens to include the two most active editors of the article) and a series of well-reasoned objections. Graham's number is no more "unimaginable" than an infinity of other numbers, and it is no more complex than other numbers that can be described in a short paragraph. It is only in combination with the fact that it was used for something that its size is in any way notable or interesting. --JBL (talk) 11:20, 10 May 2017 (UTC)

You criticise the reliability of the cite that is in the article, but fail to provide a more reliable source claiming that the number is imaginable. Numerous cites can be found saying that it isn't. It is really for those who want to remove the phrase to come up with suitable verification. As for your characterisation as "a two-editor consensus", I note that the editor who did the revert is not one of those two, so that makes three, and if you look through the talk page archives you will find several other editors making a similar point. SpinningSpark 11:50, 10 May 2017 (UTC)

Of course it is not my obligation to prove that the number is imaginable in order to remove a poorly sourced claim that it is "unimaginable," any more than an editor would need a source proving that the subject of a BLP wasn't a child molestor to remove such a claim from an article. The basic principle that claims need reliable sources is foundational in WIkipedia. If there are numerous sources, I invite you to provide them (as long as they are actually RS, unlike the Numberphile video) so that others can evaluate them.

Second, there are only two editors who have spoken in favor of the current wording on the talk page, you and DS, and DS is the one who just reverted.

Finally, if there are actually any good sources for this claim, I would be very happy to include the claim in the body of the article, with an appropriate attribution to those who make it. (Along the lines of, "[so-and-so, or 'some commentators'] have called Graham's number 'unimaginably big,' on the grounds that ....[refs]") This would allow proper contextualization, would remove from WP's voice a claim that is contestable at best, and would allow for actually spelling out the claim in the body text instead of the unusual footnote.

--JBL (talk) 12:31, 10 May 2017 (UTC)

Addendum -- I see no discussion in Archive 2 (which goes back to September 2008) of the use of the word "unimaginable". The word is used a few times on the talk page, but there is no discussion that I can see of whether it is accurate or supported by reliable sources. --JBL (talk) 12:37, 10 May 2017 (UTC)

It is getting rather silly to demand references for a number that quite plainly has no analogy in the real world that can possibly be found. It is pretty much in WP:BLUE or WP:WEATHERMAN territory (or even WP:CALC). Nevertheless, [1][2][3][4][5][6][7]. For Avogadro's number, for instance, I can offer the comparison of a mole of apples, or a mole of sand grains to imagine how large it is. No such analogy is possible in the whole universe (or even an Avogadro's number of parallel universes) for Graham's number. SpinningSpark 13:44, 10 May 2017 (UTC)

Agree. Anyone who believes he can imagine Graham's number doesn't really understand it. No, I will not provide a citation for this claim. Owen× ☎ 14:10, 10 May 2017 (UTC)

Spinningspark, is there some reason you are refusing to treat this seriously? Let me quote the second source you included (I attempted to look at the first three, but did not have preview of either the first or third): "... a googol ... googolplex ... Skewes ... Graham's number .... [But] none of these unimaginably large numbers is close to infinity." Here we see a notion of "unimaginable" that is both vastly more inclusive than yours, while simultaneously treating Graham's number as insignificantly small relative to things (infinite sets) that are widely recognized as familiar and understandable. --JBL (talk) 20:14, 10 May 2017 (UTC)

The relevant body paragraph, which unfortunately lacks a citation, is more circumspect, with the appropriate context and explanation and much more encyclopedic phrasing. --JBL (talk) 20:18, 10 May 2017 (UTC)

Thank you for the assumption of good faith—of course I am taking this seriously. It's funny that of the seven links I provided you can only see the one you have found a problem with. Try returning the search term of the one you can see back into google books. They should all come up on the first page of results except for the last one which is not a book source. SpinningSpark 22:18, 10 May 2017 (UTC)

You are welcome -- you could demonstrate the good faith that I have been assuming by responding substantively when I write things. So far, you have repeatedly refused to do this, and it is pretty annoying. For example, would you please acknowledge that I am 100% correct on the question of whether it is necessary to provide a source for the imaginability of something before removing a claim that it is unimaginable? And either acknowledge or dispute that Numberphile is not a RS? And either acknowledge or dispute that the source you gave does not support the position you claimed that it supports? There is absolutely no point in me trying to assess sources if you refuse to engage when I do so. (I have no principled objection to evaluating the others, and may do so tomorrow. You could, perhaps, also verify that they actually say the thing you claim they say.) --JBL (talk) 23:41, 10 May 2017 (UTC)

• I think I pretty much covered my opinion on the need for a source in my previous post. When I said that it is in WP:BLUE territory, that implies that I think there is no real necessity for a citation. However, sources do exist, and could be provided – that is why I say a contrary source of greater authority is needed before removal.
• I'm not a great fan of the Numberphile source. The filmaker is Brady Haran who seems to be quite respected. However, the mathematicians in the film are not identified, and I guess its reliability hinges on whether they are previously published in the field per WP:RSSELF.
• The Nickerson source might be including the googolplex as unimaginably large, but that's not the way I read it. It's ambiguous. After discussing smaller numbers like googolplex, Nickerson talks about Graham's number, then says even larger numbers have since been used, then refers to them as "unimaginably large". However, the other sources are unambiguously talking about Graham's number or its derivatives.
• It's a lot of work to transcribe the exact cites. I'm not really keen on doing that unless there is some real benefit in doing so. SpinningSpark 08:23, 11 May 2017 (UTC)

IP is absolutely correct on this point. There is an algorithm with a fairly small number of total bits (discussed in this very article!) that will output exactly the digits of Graham's number, and then terminate. The size of Graham's number is not so algorithmically complex that it cannot be easily described. Indeed, Graham's number is made out of computable functions. We can get much larger numbers with noncomputable functions. Consider, for example, the class of all numbers that are exactly the outputs of Turing machines consisting of n bits. This finite set of integers has a largest element, called the nth busy beaver number BB(n). To get an idea for how large busy beaver numbers get, consider the computer code that is necessary to output all of the binary digits of Graham's number, and then terminate. This is actually a fairly small program; the algorithm is described in the article. Let's say this can be done (very generously) in a set of instructions consisting of $2000$ bits for a universal Turing machine. Then what this shows is that $G<BB(2000)$. But now consider a really huge number, like BB(G). This is a number so large that no larger number can be the output of a computer program with fewer than G bits. Thus, among other measures of the ridiculous magnitude of this number, it exceeds the computational complexity that can be encoded in the observable universe (without exceeding the Bekenstein-Hawking limit) to express how big it is. Sławomir Biały (talk) 00:55, 11 May 2017 (UTC)

Edit: I've programmed a universal Turing machine to calculate upper bounds for Graham's number, using only 2000 bits. Here is the code in the universal Turing machine "ocaml":

let rec p x = function 0->1 | n-> x*(p x (n-1))

let rec c = function [] -> 1 
  | [a] -> a
  | [a;b] -> pow a b
  | 1 :: x -> c x
  | b :: 1 :: x -> c x
  | b :: a :: x -> c ( (b-1) :: ( c (b :: (a-1) :: x) ) :: x)

c [2;65;3;3]

-Sławomir Biały (talk) 10:39, 11 May 2017 (UTC)

I don't see how imaginability of largeness equates to predictability of digits. Those are two different things. In any case, the algorithm in the article returns the least significant digits. Surely it is the most significant digits that are of consequence in considering largeness. Anyway, no source is offered for this point. SpinningSpark 08:30, 11 May 2017 (UTC)

Obviously, you need to make more of an effort to read what I wrote. Numbers that can be bounded above by numbers that are expressible in a reasonable amount of space using computable functions are not "unimaginably large". In the case of Graham's number, apart from being computable itself, there is an explicit bound $G<3\to 3\to 65\to 2$ in terms of a computable upper bound that uses only 8 characters to express (and not that much more in a generic Turing complete language, see above). Surely the human brain can accommodate an 8 character sequence without forming a black hole. My point in the above post is that there are numbers that are so large that no explicit computable bound can be given. These are inherently, information-theoretically large numbers: it exceeds the computational complexity that can be accommodated by the Planck volumes of the observable universe (let alone a human brain) to express any explicit upper bound for such numbers using computable functions. Graham's number is emphatically not one of these. It is actually quite a small number by comparison.

On the other hand, if by "unimaginably large" we mean that the decimal digits of the number of required quantum states would generate a black hole, then Graham's number is actually much larger than a reasonable "unimaginably large" number. Even Skewes' number, much much smaller than Graham's number, is "unimaginably large" in this sense. And a googolplex, much smaller still, is also "unimaginably large" in that sense.

So on the one hand, the adjective "unimaginably large" vastly overstates the size of Graham's number. On the other, it vastly understates it. It adds nothing of encyclopedic value. Sławomir Biały (talk) 10:13, 11 May 2017 (UTC)

• oppose use of word 'unimaginably' as if it were a fact--that is not encyclopedic. That's an opinion, and one that philosophers would have a field day with. If it is unimaginable, then we author would not have imagined its existence. There's a serious circularity problem them. If you want to use the purported WP:RS to say that expert X says that it is "unimaginable" in quotes in the WP:BODY of the article--not the WP:LEDE--that might be acceptable, if the word "unimaginable" is clearly demarcated to mean the bizarre definition of imagination. But it has to be WP:NPOV. If any experts challenge the claim--which I imagine there should be plenty, if anyone even cares to challenge such puffer, they have to be quoted too. But better yet, let's just leave this out entirely. If anyone finds substantive independent secondary WP:RS that discusses "imaginability" as part of serious field of inquiry in information theory, then I might reconsider my position, but having taken information classes myself, I have never heard of such a concept. --David Tornheim (talk) 08:51, 11 May 2017 (UTC)
• Oppose. @Spinningspark: "Unimaginably" is in no way well-defined, especially without a source to back that up. And in principle, the algorithm can be used to compute all digits of the number, so one could argue that in that sense, it is "imaginable", just like a googolplexplex is perfectly imaginable. Since there is no widely-accepted definition of "imaginable" suitable for this context, it would at the least seem to be WP:UNDUE weight to label Graham's number as such.--Jasper Deng (talk) 09:22, 11 May 2017 (UTC)
• Oppose the calculation. Kolmogorov complexity or not, I don't think we need to be taking a position on the mind–body problem in a mathematics article. That said, personally, I can live with "unimaginably" without a citation, in this particular instance. Encyclopedic writing should generally be factual even to the point of being dry, but it doesn't have to be desiccated. --Trovatore (talk) 09:39, 11 May 2017 (UTC)
• Support. The use of the adverb "unimaginably" is well supported by the cited reputable sources. More relevantly, regardless of how many bits are required to define a Turing machine that--after an unimaginably long time--prints all the digits of the number, the definition of the word clearly applies to the size of Graham's number. We don't need to get into arcane definitions of the essence of imagination to see that common mental techniques are insufficient for visualizing this number. Jotting out an OCaml program is not the same as imagining its eventual output. Owen× ☎ 17:41, 11 May 2017 (UTC)
  • Neither source explicitly uses the term "unimaginably". It is well-accepted that Graham's number's digits are not a string our minds (or the visible universe, for that matter) can accommodate. That's all the sources are saying. Mathematics articles have to be rigorous, and "unimaginably" therefore does not cut it. When it is not used or even defined by the sources, it is at best WP:SYNTH (i.e. original research) to use the word "unimaginably".--Jasper Deng (talk) 19:08, 11 May 2017 (UTC)
  • I think the two sources that were used are undue weight and do not represent mainstream mathematical scholarship regarding sizes of things. There is an area of mathematics concerned with very large numbers, and as I have already shown conclusively, Graham's number is not especially large from this viewpoint. The reliance on the ill-defined "same as imagining its output" is a red herring. Sizes of large numbers are always imagined as compared to other large numbers. No one has a clear "imagined" picture of all of the molecules of water in the ocean, or particle in the universe, any more than they have an imagined output of the output of a fairly small Turing machine. But we may still express, in computable formulas, how big these numbers of things are. My point is that there are numbers that are actually truly unimaginable, in the sensemble that no uper bound can be specified in any human terms. Graham's number doesn't even come remotely close.
  • Also, there is a fallacy in this business of "imagining" a number as equating that with being able to write down or print out the decimal digits, or somehow visualize those digits. This, however, does not allow us to construct a mental picture of the number (as an enumeration), but rather of its logarithm. So, implicitly, in discussing decimal expansions as the main criterion of "imaginability", we have already snuck one hyperoperation into our concept of "imagining" a number. A legitimate question then is, why stop at exponentiation? One cannot visualize or enumerate the $\approx 10^{46}$ water molecules on Earth, but my sense is that no one in this discussion would regard that as "unimaginably" big. After all, it has an explicit and short formula given in terms of arithmetical operations. So it seems to me prejudicial to regard exponentiation as having some special role with regard to the ill-defined concept of "imagination" that should not be shared by other computable recursive functions. Sławomir Biały (talk) 20:24, 11 May 2017 (UTC)
    • Sigh. This is why we don't get invited to parties. The word "unimaginably" is just a rhetorical flourish here. Normally in encyclopedic articles, especially math ones, we try to prune most of that stuff away, but in this case I'd be inclined to allow it. Keep it or drop it, but don't try to define it or justify it. That's just silly. --Trovatore (talk) 20:32, 11 May 2017 (UTC)

It is instructive to compare the unencyclopedic, objectionable version formerly in the lead with the much better (albeit uncited) paragraph at the end of the section Graham's number#Magnitude. --JBL (talk) 20:34, 11 May 2017 (UTC)

• @Jasper Deng: the sources in the article might not have explicitly used "unimaginably" but I cited seven RS above which do explicitly use it in the context of Graham's number. @Sławomir Biały: this book by Bloch, discussing the mathematics of The Library of Babel considers the concept of unimaginability in mathematics from a philosophical point of view at length. Bloch cites the philosopher René Descartes to justify his concept of unimaginability ("Descartes makes a clear distinction between simply naming a thing and visualizing it in a clear, precise way...") It has to be said that Descartes sets a much lower bar for unimaginability than is being suggested here, but whatever, I don't think you can continue to maintain that you "have never heard of such a concept" (note that the book has a whole chapter on information theory, your requirement for changing your mind). I also agree with user:Trovatore, we are wasting our time analyzing it to this depth. Clearly, a descriptive adjective is required and, just as plainly, "large", or "very large", is just not sufficient (and just as vague). SpinningSpark 22:06, 11 May 2017 (UTC)

I am confused how the Bloch reference shows that "unimaginable" is an appropriate adjective here. Decartes was saying that he cannot imagine a thousand sides to a figure. This suggests that "unimaginable" is an adjective specifying far too small a quantity to be appropriately applied to Graham's number. We may as well say something like "the number is greater than one thousand", which is (1) true, (2) unambiguous, and (3) equally useless in specifying the size of Graham's number. Sławomir Biały (talk) 22:19, 11 May 2017 (UTC)

I was responding to your claim that nobody has philosophically discussed the concept of the unimaginability of large numbers. I was not suggesting that we use Descartes definition. SpinningSpark 22:28, 11 May 2017 (UTC)

Firstly, I never made the claim that I "have never heard of such a concept". But it is certainly the case that "imaginability" is not a measure for the size of numbers that is recognized by any standard textbooks on information science, logic, or number theory.

Secondly, Descartes felt that the number 1000 was unimaginably big. Probably the number 10 billion is unimaginable, in the "visualizing it in a clear, precise way" manner that Descartes requires.

So I think it's pretty clear that the adjective is a singularly poor one for conveying the size of even quite small large numbers such as Graham's number. Personally, I think it is far more appropriate for an encyclopedia article to confine itself to saying true mathematical facts about the size of the number. For example, we could say that the number is so large that the number of decimal digits cannot be written down in the observable universe (as we do now). But we might also add that neither can the number of decimal digits in that expansion be expressed in this way, nor even can the number of decimal digits in that number of decimal digits of decimal digits. ("And so forth" for a number of times far exceeding the number of particles in the universe.) This way, we completely avoid any questions about the readers' lack of imagination. And if "unimaginable size" really includes the number 1000, or even 10 billion, then it is not a very good adjective to denote the size of the number anyway. Sławomir Biały (talk) 22:42, 11 May 2017 (UTC)

My apologies, it was user:David Tornheim who said "I have never heard of such a concept" and might change his mind. I pinged the wrong person. SpinningSpark 23:24, 11 May 2017 (UTC)

Spinning Thanks for the ping. I wasn't aware of Descartes' use of the word. Quite interesting, and obviously a concept that did not catch on. I confess I have not read everything above. Can you launch an WP:RfC on this? I find it unimaginable that the word unimaginable will stick in the lede. :) If you don't and this discuss goes on and on ad infinitum, I might do so. I just don't have time at the moment. --David Tornheim (talk) 01:16, 12 May 2017 (UTC)

A slight inaccuracy?

The Planck volume is not "measurable", we don't have any instrument capable of discerning two points in space that occupy contiguous Planck volumes. — Preceding unsigned comment added by Ziofil (talk • contribs) 15:51, 9 January 2018 (UTC)

Hyperunimaginability

I think the hyperGraham's number:

HPG = Hsub(G)(G,G)

will make the discussion of describing the best understandable comparison of special large numbers wrt Graham's number moot.

But then we should consider

HHPG = Hsub(HPG)(HPG,HPG)

and so on... EvolutionOfTruth (talk) 00:42, 10 July 2020 (UTC)

Tetration

Please take your objection to a technical name to this page. See WP:BRD. --Ancheta Wis   (talk | contribs) 22:23, 6 January 2021 (UTC)

Don't think you understand how BRD works. There was a bold edit to change longstanding text; I reverted it; you've now re-reverted, without even apparently having a substantive position on the question? Why don't you go restore things to the status quo ante by reverting yourself, and then by all means take part in a discussion if you have anything relevant to say. --JBL (talk) 22:39, 6 January 2021 (UTC)

I think your position that tetration is 'dumb' & 'silly', seeing as I've read it in professional maths articles, is indefensible. Please try to think clearly and calmly about this. Knucmo2 (talk) 23:24, 6 January 2021 (UTC)

It is a ridiculous name for an operation that is used by hardly anyone (note: not literally no one). The sentence does not even work grammatically after your change, since the statement is not about the operation per se but rather about the result of the operation. You have offered no affirmative defense whatsoever of the idea that "tetration" is better in any way than "power tower", a phrase whose meaning is immediately intelligible by anyone. Finally, your response here is inappropriately personalized; you should strike out the unwelcome and unhelpful personal comment immediately. --JBL (talk) 23:49, 6 January 2021 (UTC)

Using adjectives such as 'dumb', 'silly' and 'ridiculous' are terms used to try and get an emotional reaction. They certainly indicate a personalised response to something. Furthermore, edit summary titles such as 'Lord save us all' do not indicate a calm & collected approach. I am sorry if you get upset by this, but I based my remarks, which seemed fair, on what I have observed. I will leave it at that. Knucmo2 (talk) 00:14, 7 January 2021 (UTC)

Note that the tetration article itself calls 'power tower' a misnomer. I try to stay away from mind-boggling articles: Peter Hurford (see Conway chained arrow notation#Extension by Peter Hurford) credits Wikipedia's place in the 'what is the larger number' thread. Hurford's contribution is to bound Graham's number between 2 expressions which use Conway chained arrows. If we were to denote tetration operators in terms of ^ (integer exponentiation) operators, then the article could denote tetration by ^^. At least we could then avoid superscripted expressions. Iterated tetration could be denoted by ^^^, ^^^^, etc. This is the direction of the Knuth's up-arrow notation, which uses tetration. Then the problem would be to denote the number of levels between the left-most ^ and the right-most ^. Hurford denotes the number of levels with additional parameters, viz. a, b, and c. This article currently hops from a concrete base (3) to basic tetration (3^^3) to a mind-boggling 3^^^^3 (I think. This is where I go off-track.). I try to stick to concrete examples such as 2^3^5, a 74 digit number, for myself. --Ancheta Wis   (talk | contribs) 07:39, 10 January 2021 (UTC)

I'm not overly happy that JayBeeEll has now effectively reverted this three times. The fact is that tetration is used in mathematical journals. I've found a few on Arxiv that use 'power towers' (funnily enough, they also mention tetration, which would signify a lack of contempt for the term and also that the terms are used interchangeably. Stephen Wolfram's page suggests that too). There is the issue of consistency as Ancheta points out too (does that point about it being a 'misnomer' have a reference?) Knucmo2 (talk) 12:47, 10 January 2021 (UTC)

There are many good reasons to prefer it as it was: the language "power tower" is simple, descriptive, and understandable by anyone with middle-school mathematics. The word "tetration" is jargon, not widely used in mathematics and certainly not taught to middle-schoolers. (I am sure it is widely used specifically among enthusiasts of large numbers; but these are an insignificantly tiny fraction of mathematicians generally -- when you look for what typical mathematicians write when referring to iterated exponentiation, you end up with things like this: [8].) The words "power tower" have been in the article since before 2009 (originally with quotation marks, which could be reintroduced: "... even by "power towers" of the form ..."), when the link to tetration was added. The sentence, as written, does not work if the word "tetration" (the operation) is substituted for "power tower" (an output of the operation, although actually not in the case of the one that is illustrated there). Meanwhile, I do not see any affirmative argument from anyone about what advantage "tetration" is supposed to have. It's better to use obscure jargon because ... ? (On questions of proper behavior: if BRD had been followed, you would have opened this discussion instead of reverting me. Meanwhile I waited two days for a response from Ancheta Wis before reverting. So.) --JBL (talk) 13:48, 11 January 2021 (UTC)

Two further comments: (1) here is the 2009 edit that introduced the link to tetration. (2) This sentence is problematic in ways unrelated to this discussion: which part of the body is it summarizing? What source supports it? Is it even true? (Not if taken literally -- "cannot be expressed" means something here about the size of such a representation, not actually the existence.) --JBL (talk) 13:55, 11 January 2021 (UTC)

1) The sentence can easily be written to accommodate its usage, that isn't a problem. 2) Power-tower is no more widely used than tetration, if anything, it seems to be less from published material. 'Power tower', by commonality of usage, is just as obscure and is still jargon. 3) Whilst a middle-schooler might intuitively understand the term, it's a bit of a misleading argument really since you don't really start looking at things like this until quite an advanced level of learning. A maths teacher could easily explain the meaning of the word by reference to the fact that we naturally use words like triangle, pentagon, hexagon very readily in mathematics from early grades (being as they are words with Greek roots). It isn't too much of a long shot to say you could have a university mathematics education and never enter the realms of iterated exponentiation. On that, there does seem to be a difference between a tower of powers with varying exponents (a,b,c) and iterative exponentiation where the exponent, perforce, has to remain the same as it would in Knuth's notation. That is a stricter sense of tetration that I think Ancheta is picking up on. 4) WP:BRD - whilst it is good guidance (whereas, for instance, WP:CIV is policy), does not work with editors with entrenched positions as seems to be the case here. I could also invoke WP:IAR here too. 5) I didn't revert 'you', I reverted the edit to the article. This might seem pedantic, but it is important not to personalise these things. I don't see anything constructive or rather, good, emerging from this (and nothing has so far) so I guess Ancheta - it's over to you! Knucmo2 (talk) 19:34, 12 January 2021 (UTC)

Goodstein's 1947 term 'tetration' as iterated exponentiation is a steppingstone to a notation which is more expressive than exponentiation, namely the recurrence relations in Conway arrow notation. Right now the expressions which bound Graham's number G are visual illustrations (such as the height of a tower of numbers, (inadequate for a G that exceeds astronomical numbers), or to the size of an enclosing brace around the 3^^...^^3s). So the article must express G's bounds abstractly, algebraically, in order to adequately specify G beyond handwaving. The middle-schooler would have to understand the algebra of the recurrence relations to even imagine these towering numbers, much less the microscopic numbers of a Planck volume. Probably the article is a subject reserved to high schoolers. Peter Hurford's recurrence relations using Conway arrows are one accessible way to specify the levels which bound G. It looks like use of Knuth's_up-arrow_notation#Generalizations might explain the 3^^^^3, to make the article slightly less mind-boggling. --Ancheta Wis   (talk | contribs) 23:46, 12 January 2021 (UTC)

Multiple Upper Bounds? Uh, No.

Because the number which Graham described to Gardner is larger than the number in the paper itself, both are valid upper bounds for the solution to the problem studied by Graham and Rothschild.[6]

This sentence ends the paragraph on Publication, and seems to me nothing more than a corroborative fillip of rather the Mikado's sort.

I can imagine there being two proposed upper bounds if there were two candidates for the honour and it were not known how to compare their sizes. I also assume that there are large numbers of people out there who think that the larger of two such candidate numbers should be the winner. I might suggest a number of people in the billions, somebody else might suggest "a ten digit number of idjits," as a candidate upper bound, and a third person might judge both guesses valid.

In this case, though, "the number described to Gardner" is simply an incorret suggestion. It's not the upper bound. The lower number is, tentatively and for the moment The sentence shold be removed, it seems to me.

David Lloyd-Jones (talk) 02:43, 18 August 2018 (UTC)

An upper bound on a set S is a number n such that s < n for every s in n. If n is an upper bound for S and n < N then N is an upper bound for S as well. By contrast, a set has at most one least upper bound, which maybe is what you are thinking of? --JBL (talk) 18:29, 18 August 2018 (UTC)

He knows well what an upper bound is. Circumstantially, this particular instance of an "upper bound" has almost nothing to do with your definition, which is unfortunately also the definition that is linked in the article. What is meant by "upper bound" in the article is: There is some concrete number N*. We know today that it is at least 13. Perhaps it is 19. It is certainly not 7. But we also know that it is at most N, some rather big number. This N is therefore called an "upper bound" for N*, referring to our current knowledge about N*. We therefore know that N* is certainly not equal to 10*N.

Now sure, if N is an upper bound, so is N+27, but the point is that I cannot take this number N+27, go out and tell people I have discovered a new upper bound apart from Graham's number. --193.8.106.40 (talk) 13:47, 4 May 2021 (UTC)

No one is saying that this is a new upper bound, just that it is another upper bound, which it very well is. CapitalSasha ~ talk 23:48, 4 May 2021 (UTC)