Breadcrumb
Unprovable Theorems
Tue 15 December 2009, 14:00
Harvey Friedman
Ohio State University
Organiser: Guy Nason
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
called The Upper Shift Fixed Point Theorem. This involves only the order and the +1 function on the 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.