28 - Speed-up

Summary

I can't resist briefly noting here some rather striking results about the length of proofs. And it is indeed worth getting to understand what the theorems say, even if some of the proofs in this chapter are perhaps just for enthusiasts.

The length of proofs

We might expect that, as a general tendency, the longer a wff, the longer its proof (if it has one). But can there be any tidy order in this relationship in the case of nice theories?

We will say that a proof for φ is f-bounded, for a given function f, if the proof's g.n. is less than f(φ). Then it would indeed be rather tidy if, for some theory T, there were some corresponding p.r. function fT which puts a general bound on the size of T-proofs – meaning that, for any φ, if it is a provable at all, it has an fT -bounded proof. However, unfortunately,

Theorem 28.1 If T is nice theory, then for any p.r. function f, there is a provable wff φ which has no f-bounded proof.

Proof sketch Suppose the theorem is false. That is, suppose that there is a p.r. bounding function fT such that for any φ, if it is T-provable at all, it has a f_T-bounded proof. Then there would be a p.r. procedure for testing whether φ has a proof in T. Just calculate f_T(φ), and do a bounded search using a ‘for’ loop to run through all the possible proofs up to that size to see if one of them is in fact a proof of φ.