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There seems to have some interest, judging by recent questions, in numbers that are very, very large (but finite). Some examples that I am thinking of include

  • Graham's number
  • The Ackermann numbers (okay, after the first 2 or 3, say)

There's also some interest in various notations for denoting these large numbers

  • Knuth's up-arrow notation
  • Ackermann functions
  • Steinhaus polygon notation
  • Conway chained-arrows

I wonder if there should be a tag that aggregates all of the above, since they are, in many ways, related (having to do with recursion constructions and what not).

Question

Should there be a tag that captures the idea of "very large numbers and their notations"? And if yes, what should the tag name be?

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  • $\begingroup$ «Large numbers» sounds like somewhat... superficial grouping. It seems, mathematically (at least some of) these questions about... recursion maybe? $\endgroup$ – Grigory M Jan 28 '14 at 11:43
  • $\begingroup$ @GrigoryM I disagree. Very large numbers in the sense of this question are certainly a mathematical topic to themselves. However, I cannot think of a tag that would likely be used correctly, so I am not in favour with a tag like "very large numbers". $\endgroup$ – Phira Jan 28 '14 at 12:03
  • $\begingroup$ I am also interested in these very large numbers. In particular, I try to get a feeling for the magnitude of them. For example, does Steinhaus or Knuth's arrow up show a bigger growth of the numbers. It is difficult to suggest a tag, but what about "extremely-large-numbers" ? $\endgroup$ – Peter Jan 28 '14 at 12:17
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    $\begingroup$ There already exists (big-numbers) tag. But I am not sure whether the usage of this tag is along the lines you intended for "very large numbers". $\endgroup$ – Martin Sleziak Jan 28 '14 at 13:56
  • $\begingroup$ @Martin: we can always take inspiration from them crazy set theorists. :-) I am thinking mostly along numbers so large that they do not admit any reasonable representation using the familiar operations of addition, multiplication, and exponentiation. $\endgroup$ – Willie Wong Jan 28 '14 at 14:05
  • $\begingroup$ @Phira: do I understand that you are in favour of such a tag provided a suitably descriptive tag name can be found? $\endgroup$ – Willie Wong Jan 28 '14 at 14:06
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    $\begingroup$ Incidentally, even though I don't have a tag name, I do have a tag wiki excerpt available: "This tag is for numbers that are big. Really big. You just won't believe how vastly, hugely, mindbogglingly big they are." :-) $\endgroup$ – Willie Wong Jan 28 '14 at 14:09
  • $\begingroup$ There is a wiki devoted to this subject, but somehow I doubt "googology" is sufficiently self-explanatory. $\endgroup$ – user642796 Jan 28 '14 at 14:28
  • $\begingroup$ @WillieWong Yes, my pessimistic view of the consequences is my only objection. $\endgroup$ – Phira Jan 28 '14 at 17:45
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    $\begingroup$ Well, since natural numbers are cardinals, won't that make them "large cardinals"? So I suppose that these numbers of yours are going to encode some proof of the consistency of large cardinals somehow? :) $\endgroup$ – Asaf Karagila Jan 28 '14 at 19:39
  • $\begingroup$ @WillieWong Than it would be specifically aboput numbers $\ge42$. $\endgroup$ – Hagen von Eitzen Jan 29 '14 at 13:13
  • $\begingroup$ Almost all natural numbers are too large to comprehend, yet for any number, most natural numbers are larger. $\endgroup$ – PyRulez Jan 29 '14 at 23:18
  • $\begingroup$ Also see googology.wikia.com/wiki/Googology_Wiki, a wiki for so called "larger" numbers. $\endgroup$ – PyRulez Jan 29 '14 at 23:18
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    $\begingroup$ @PyRulez: that's the same link as Arthur Fischer posted yesterday. $\endgroup$ – Willie Wong Jan 30 '14 at 7:34
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How about "fast-growing functions"? The examples of numbers that you mention are obtained by considering functions from well established fast-growing hierarchies. There are natural questions about descriptions and notations, both for large numbers and for small countable ordinals, that would naturally fit here as well, and I believe people interested in "very large numbers" would also be interested in these closely related topics.

The term is used in the literature, it is more descriptive, and more likely to be something people search for than "very large numbers". Also, I think it would be more visible than if instead we talk of "fast-growing hierarchies", which (at least to me) would seem to indicate a narrower, more technical focus.

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    $\begingroup$ My advisor told me a few days ago, that in the height of the 1984 inflation in Israel, he was invited to a news panel as "an expert of large numbers". :-) $\endgroup$ – Asaf Karagila Feb 8 '14 at 19:17
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    $\begingroup$ I have been asked to participate in projects involving huge databases for similar reasons. $\endgroup$ – Andrés E. Caicedo Feb 8 '14 at 19:23
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    $\begingroup$ Wait until you're asked to referee games, or join theological panels as an expert on "determinacy". ;-) $\endgroup$ – Asaf Karagila Feb 8 '14 at 19:34
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There already exists a tag called . (At the moment, it contains 39 questions. The tag-wiki for this tag is empty, so the usage of this tag is worth clarifying.)

For example, several questions on Graham's numbers or on Knuth's up-arrow notation can be found in this tag.

Maybe we could keep using the tag, that we already have.

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