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May 10th, 2015, 08:46 AM   #1
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Are large numbers definable?

So I've been looking up large numbers lately and my question is, can we even define some of those? A lot of the information I read looking them up seems pretty inconsistent. For example, I remember seeing a page that prints out a googolplex on the screen, while there are supposedly more digits in it than atoms in the universe.
Graham's number is another famous large number example, which seems to be defined/calculated differently based on where I look it up, same with larger numbers. Are large numbers simply difficult to comprehend? More importantly, do we know, or can we prove that numbers go on indefinitely? We can also ask questions like "does 2+2 always equal 4?" but that goes beyond mathematical theory I think.
Is there a purpose to defining these types of large numbers? Take Graham's number for example, apparently it's defined by first having towers of exponents, than repeating those over and over, than repeating that over and over many many times and so on. Do these numbers serve a practical purpose or do people make them for the sake of having large numbers named?
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May 10th, 2015, 01:34 PM   #2
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Your question seems more philosophical than mathematical. What is the point you are trying to make?

Numbers go on indefinitely? Yes - proof by contradiction - if you think it stops, add 1 to the last one.

2+2=4 comes out of an axiom system for arithmetic.
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May 10th, 2015, 03:39 PM   #3
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I don't see any reason to "throw away" large numbers. In Physics, though, we tend to scale things so all the action takes place about the origin. This is true for ridiculously small numbers as well.

-Dan
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May 11th, 2015, 08:15 AM   #4
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Not really throw them away, but define them more clearly I guess.
At least it seems like really large numbers are defined differently sometimes.
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May 11th, 2015, 09:00 AM   #5
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If you accept that there is some number $N$ such that all real numbers $x>N$ are "large" (maybe N is 100 or googolplex or Graham's number), then there are large numbers which are not definable. In fact, definable numbers are quite rare -- there are only countably many out of an uncountable sea of real numbers.

But if you accept that, say, all numbers between 0 and 1 are "small", then the same is true of small numbers. So it's not so much that some numbers are so large that they can't be defined, but rather that most numbers can't be defined at all, small or large.

On the flip side, for any fixed $N$ there is a real number $x>N$ which can be defined -- say, $\lfloor N\rfloor+1.$ Certainly googolplex and Graham's number are perfectly well-defined.
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May 11th, 2015, 10:08 AM   #6
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So can one apply Cantor's diagonal procedure to the definable reals between 0 and 1?
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May 11th, 2015, 10:36 AM   #7
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So can one apply Cantor's diagonal procedure to the definable reals between 0 and 1?
Sure, and it will give a non-definable real in that range.
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May 11th, 2015, 06:16 PM   #8
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I have no problem with large numbers.

After all, most integers have billions of digits.

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May 11th, 2015, 09:40 PM   #9
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Sure, and it will give a non-definable real in that range.
Isn't it definable as the real generated by the diagonal argument using the given list of reals?
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May 12th, 2015, 06:34 PM   #10
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Isn't it definable as the real generated by the diagonal argument using the given list of reals?
The point of the diagonal argument is that the list is supposed to be complete. You then create something which should be on the list but isn't - contradiction.
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