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 May 10th, 2015, 09:46 AM #1 Newbie   Joined: May 2015 From: US Posts: 4 Thanks: 0 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?
 May 10th, 2015, 02:34 PM #2 Global Moderator   Joined: May 2007 Posts: 6,438 Thanks: 562 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.
 May 10th, 2015, 04:39 PM #3 Math Team     Joined: May 2013 From: The Astral plane Posts: 1,669 Thanks: 657 Math Focus: Wibbly wobbly timey-wimey stuff. 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
 May 11th, 2015, 09:15 AM #4 Newbie   Joined: May 2015 From: US Posts: 4 Thanks: 0 Not really throw them away, but define them more clearly I guess. At least it seems like really large numbers are defined differently sometimes.
 May 11th, 2015, 10:00 AM #5 Global Moderator     Joined: Nov 2006 From: UTC -5 Posts: 16,046 Thanks: 937 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms 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.
 May 11th, 2015, 11:08 AM #6 Global Moderator   Joined: Dec 2006 Posts: 18,595 Thanks: 1493 So can one apply Cantor's diagonal procedure to the definable reals between 0 and 1?
May 11th, 2015, 11:36 AM   #7
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Quote:
 Originally Posted by skipjack 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.

 May 11th, 2015, 07:16 PM #8 Math Team   Joined: Dec 2006 From: Lexington, MA Posts: 3,267 Thanks: 407 I have no problem with large numbers. After all, most integers have billions of digits. Thanks from CRGreathouse
May 11th, 2015, 10:40 PM   #9
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Quote:
 Originally Posted by CRGreathouse 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?

May 12th, 2015, 07:34 PM   #10
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Quote:
 Originally Posted by v8archie 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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