Concrete Mathematical Incompleteness

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) - and that has been proved using standard extensions of ZFC generally adopted in the mathematical logic community. The highlight of the talk is the presentation of a new unprovable theorem concerning the structure of kernels in discrete and finite directed graphs. We first review some previous examples of unprovable theorems, as time permits. 1-5 are unprovable in the weaker sense that any 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.