# Unprovability

From: Lee Corbin (lcorbin@ricochet.net)
Date: Wed Apr 04 2001 - 00:38:26 MDT

Mikael Johansson wrote

> His name is Pál Erdös -- not Paul Eotvos; I presume?

Yes. That's who I meant when I referred to how he believed
that "God's Book" contained all, and only, the most elegant
proofs. Erdos was always saying "That's in The Book!"
when shown a really exquisite proof. I was spoofing when
I suggested that God's Book, in it's appendix, might also have
all the true relationships, with an asterisk next to the
ones that had proofs, (and two asterisks next to the ones
that could be proved by pre-singularity humans!), no matter
how ugly the proofs.

> ...if it is unprovable whether there exists infinitely many
> prime pairs, then the existence of an upper bound on the
> prime pairs is unprovable, and thus -- perchance -- one might
> argue that the unprovability proves the theorem.

That's great! Whether you are right or not! It makes my head
hurt, and that's the purpose of my little story about the
mathematician at the party, namely as an aid to keeping clear
what I know about provability and unprovability.

Let me take a swing: Okay, so the student says that he
has a proof that (the existence of infinitely many) is
unprovable. So he has a proof (in his trembling fingers)
that it's unprovable whether there is an upper bound.
That means that for each integer (beyond the small ones,
of course) we cannot prove whether or not it is an upper
bound.

Interestingly, suppose that his proof proves that for each
integer N greater than 3^163^641^6969 + 86^86 + 42 you
can neither prove nor disprove (i.e. such proofs do not
exist) that N is an upper bound. Then that would mean
that any pair greater than that would have to be
composite (because we can always, brutally if need be,
establish whether or not a given number is prime), and
as you say, that would prove the finitetude of the prime
pairs.

It doesn't help any if his paper merely proves that there
exists some number Z such that you can neither prove nor
disprove that any number greater than Z is an upper bound.
Your observation kicks in once more. The student has been

But what if his paper just proves that you cannot prove that any
number greater than 42 is an upper bound? (Slightly different!)
Just because we can't prove it, it may nonetheless be the case
that 3^163^641^6969 + 86^86 + 42 is in fact an upper bound
(although we could never possibly learn that). Then we would
have to hail the student and his proof after all.

Thanks,
Lee

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