User blog comment:XXxKennyDashxXx/A NEW NOTATION! HBAN./@comment-25291678-20141227153743/@comment-1605058-20141227165018

Suppose we can describe a Turing machine which looks through all valid ZFC proofs seeking for a proof of contradiction, and say that this machine has N states. Then I claim that ZFC cannot prove, for any Turing machine (or a C program, or an equivalent model), that this machine (program) halts with more than S(N) symbols. Indeed, if it could prove that machine M halts with K>S(N) ones, then we could simulate contradiction-seeking machine for K steps. If it didn't halt, then it will never halt, by definition of S(N). But this way ZFC could prove its consistency, contradicting Godel's incompleteness.

It's not hard to believe that N is quite small, say smaller than 10^100, and that BIG FOOT is greater than S(10^100) (we can define Turing machines in FOOT), so it shows that, for any "reasonable" formalization of "computable integer" we will never be able to prove any such integer to be bigger than BIG FOOT.

(of course, if one gives notion of "computable integer" too much freedom, the above could easily not be the case, which is why I mention "reasonable")