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That might be a silly question, but here goes: I see a lot of "big numbers" in physics, such as the size of the state space of all the particles in the visible Universe, and those numbers can be written down, like $10^{10,000}$, or whatever that particular value turns out to be. Maybe there comes a time where we couldn't even write down a large number, because we wouldn't have enough atoms in the Universe. At that point, we might resort to a program or a sentence to describe how to compute the number that we could never actually write down. Then maybe we could get to an even larger number that can't even be described with any program or sentence, not even in principle? Is there such a number?

Frank
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    I think you would be interested in an old article, "Who can name the biggest number?". It discusses a lot of these concepts. – Jair Taylor Apr 28 '15 at 19:02
  • Yes - but I want the number after the last number we can name :-) – Frank Apr 28 '15 at 19:05
  • If you keep reading, he discusses the "Busy Beaver Numbers" - a sequence of numbers that grows more quickly than anything describable by a computer, which is something like what you are looking for I think. – Jair Taylor Apr 28 '15 at 19:08
  • I'm starting to think the number I'm looking for does NOT exist. The reason is that it is NOT in the universe at all, since it's not even in my mind, and there is nothing but the universe... wait! did I just say that? Is this still maths? – Frank Apr 28 '15 at 19:12
  • I think there's a googology wiki somewhere. Googology is the study of large numbers. – Akiva Weinberger Apr 28 '15 at 19:20
  • Look up Rayo's number. It's the smallest number that can't be written out (in the language of set theory) with a google symbols or less. – Akiva Weinberger Apr 28 '15 at 19:21
  • Smallish - they stopped at a google symbols or less, completely arbitrary, small limit ;-) – Frank Apr 28 '15 at 19:30
  • googol (the number) is not the same as Google (the search engine). – JRN Apr 29 '15 at 00:46
  • In my limited experience, if you have even an extremely large maximum number, the theory of basic arithmetic on a finite set (where max + 1 is undefined) gets really ugly really fast. Infinity really simplifies even basic arithmetic in this sense. – Dan Christensen Apr 29 '15 at 03:37

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If you have a fixed set of symbols, and set a bound on how many characters you are allowed, then you can only distinguish a finite number of numbers. Since the observable universe is finite, you can only use atoms in the universe to distinguish a finite number of numbers. So, yes, there are an infinite number of numbers that cannot be explicitly specified in a way that distinguishes them from all other numbers. And we're not even talking about classes of infinity here, just finite numbers that there aren't enough atoms to describe.

There are paradoxes about writing down the smallest such number, and that's an interesting topic. But the fact is, most (all except for finitely many) numbers are just too big to write down.

Mark Foskey
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No,It is not possible. Infinity or infinite is nothing but a concept. It can be explainable is terms of Mathematics and may be use in diversified field of science. If we consider infinity mathematically, we find it as endless process, of larger than large number and smaller than small number ( for $-\infty$ or negative infinity). We may find different application of infinity in particles of physics. In case of universe we also find it's application but we may consider universe as finite, if we count it by a fixed large quantity. But here is the question, what will be fixed value of this large quantity? Which parameter we take? If we don't know this fixed quantity of large number, we consider it as it goes to infinite extend. So it is relatively defined. If we consider countable number of stars in universe (not in our planetary system ) it will goes to infinite extend, if we don't know fixed quantity. As if we consider any finite dimensional system say dim$(n)$ i.e. $n$ dimensional in any space say vector space. If we take as $n\to\infty$ then we will find infinite dimensional system in vector space. So infinity or infinite is relatively defined concept, not absolute.

dantopa
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wings of mathematics by Tanu S
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