April 13
Ryan Utke,
CUNY
An Incompleteness in Peano Arithmetic
In 1931, Godel proved his incompleteness theorem, that any recursive axiomatization of arithmetic is incomplete. As a consequence, there exist true statements about the natural numbers which cannot be proven from the usual Peano axioms. However, the first examples of such statements relied on coding metamathematical concepts such as consistency or provability, and for many decades it was unknown whether any 'natural' statements in arithmetic are true but unprovable. In the 1970s, such a statement was given by Paris and Harrington, whose argument we present. In particular, we prove (in ZFC) that a variant of the finite Ramsey theorem is true but implies the consistency of PA, hence cannot be proven in PA.