[Doctor Stochastic]

Note that in your comments, you have proved that there are an infinite number of primes, not that there is an infinitely large prime.

(It's possible that) one may show that for any N, there is a proof that requires N+1 steps, but that doesn't mean that there is a proof that requires infinitely many steps. To show that, one must exhibit a statement that, for any N, cannot be proved in N or fewer steps.

[LibWhacker]

Yep, I made two mistakes here. First, I thought it was trivial and intuitively obvious that if there were proofs of any given length N, then necessarily there were infinitely long proofs. Second, I thought that by showing there were infinitely long proofs that that would disprove the antecedent of the first, because the idea of an infinitely long proof is meaningless (wrong again!).

But in trying to prove it, I found out in doing a little research that not only are there proofs that are infinitely long, their existence contradicts nothing! Some mathematicians seem quit comfortable with the notion. This is all covered in a field called "proof theory," to which I had no exposure in grad school, and I gather is an area of mathematical logic.

It still seems very strange to me, though, and by "proof" they must be working with some generalized definition of "proof." Heck, I can even think of a specific example of a countably infinite proof now, though I have no idea if this is what they are talking about, but I think it must be. For example, take Catalan's conjecture that says 8 and 9 (2³ and 3²) are the only consecutive powers in Z⁺:

Proof
1 and 2 aren't consecutive powers
2 and 3 aren't consecutive powers
3 and 4 aren't consecutive powers
4 and 5 aren't consecutive powers
5 and 6 aren't consecutive powers
6 and 7 aren't consecutive powers
7 and 8 aren't consecutive powers
8 and 9 ARE consecutive powers
9 and 10 aren't consecutive powers
10 and 11 aren't consecutive powers
11 and 12 aren't consecutive powers
. . .
And so on, forever. Q.E.D.

This "proof" has a countably infinite number of lines and if such infinitely long proofs are allowed, is in fact a "proof" since it exhausts all consecutive pairs -- PROVIDING, of course, that the conjecture is true (I suppose I should've used an example whose truth is known, but I'm too lazy now to go back and change it).

I even found references to artificially expanding any finite proof so that it has an infinite number of lines, and now that I've read a little bit about proof theory, I wouldn't doubt at all that it's possible and wager that if a person tried, it wouldn't be too hard to come up with an example of that also.

Finally, I found a claim that there exist proofs that are not only infinitely long, but have infinite logical "depth;" i.e., that are hopelessly beyond man's understanding. This is bad news, imo. Never dreamed there were results like this out there. Nice talking to you guys.