Harvey Friedman

Abstract:   An unprovable theorem is a theorem about basic mathematical objects that can only be proved using more than the usual axioms for mathematics (ZFC = Zermelo Frankel set theory with the Axiom of Choice). The highlight of the talk is the presentation of a new unprovable theorem that involves only the usual ordering of the rationals, vectors of rationals, and the +1 function on rationals.

We first review some previous examples of unprovable theorems in the weaker sense that the proof demonstrably requires some use of logical principles transcendental to the problem statement. These previous contexts include

1. Patterns in finite sequences from a finite alphabet.
2. Pointwise continuous embeddings between countable sets of reals (or, more concretely, rationals).
3. Relations between f(n_1,...,n_k) and f(n_2,...,n_k+1).
4. Homeomorphic embeddings between finite trees.
5. Borel sets in the plane and graphs of one dimensional Borel functions.
6. Boolean relations between sets of integers and their images under integer functions.