Talk:Graham's number

Description of Graham's Number in the intro

I have something I'd like to get a little clarity on. The intro contains this wonderful description of the scale of Graham's Number:

...it is so large that the observable universe is far too small to contain an ordinary digital representation of Graham's number, assuming that each digit occupies one Planck volume, possibly the smallest measurable space. But even the number of digits in this digital representation of Graham's number would itself be a number so large that its digital representation cannot be represented in the observable universe. Nor even can the number of digits of that number. And so forth, for a number of times far exceeding the total number of particles in the observable universe.

I absolutely love this description, but I have a question on one aspect of it. Throughout, the description uses the concept of the number of Planck volumes in the observable universe, which is fine... but by the end, it switches to "a number... far exceeding the total number of particles in the observable universe".

This switch from Planck volumes to particles confuses me. What I'd like to know specifically is this: is

a number... far exceeding the total number of particles in the observable universe

larger than the number that can be expressed with the number of Planck volumes in the observable universe?

Or to put it another way: is the description intending to say that you could, in fact, express that number with the number of Planck volumes in the observable universe?

Hawthornbunny (talk) 19:42, 6 November 2017 (UTC)

Hi there,

No you couldn't. Graham's number is far bigger than the number you would get by calculating that number, whether you take the smallest unit of volume as being that of the smallest particle yet known or Planck's volume itself. In fact it is bigger than anyone can begin to imagine. Meltingpot (talk) 22:25, 20 May 2022 (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)

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)

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: [1].) 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)

Orders of magnitude larger

@Yekshemesh: re this edit I don't think that "Graham's number is a great many orders of magnitude larger than other large numbers such as Skewes' number and Moser's number" actually is does "better captures how big this number is" as your edit summary claimed. Graham's number cannot even sensibly be expressed as an order of magnitude, or even a power tower of orders of magnitude as those other numbers can. It's actually a quite inadequate way of capturing the scale. SpinningSpark 19:49, 20 October 2021 (UTC)

@Spinningspark: Sure. Yekshemesh (talk) 19:52, 20 October 2021 (UTC)