Forcing over arithmetic: a second-order approach

by kamerynwilliams

This will the inaugural talk in the CUNY Models of Peano Arithmetic seminar for the Spring 2018 semester. It will be Wednesday, February 7. See the temporary webpage for the seminar for more precise details.

Continuing a theme of previous talks in this seminar, I will talk about forcing over models of arithmetic. I will present a framework for forcing over models of \mathsf{ACA}_0, generalizing the approach (as seen in e.g. Kossak and Schmerl’s book) of looking at definable sets over a model of PA. This is analogous to the approach within set theory of developing class forcing in GBC, rather than using definable classes over ZFC. The main result I will present is that forcing preserves the axioms of \mathsf{ACA}_0 and, indeed, \Pi^1_k\text{-}\mathsf{CA}.

This is part of joint work with Corey Switzer about generalizing Kossak and Schmerl’s results about perfect generics. Stay tuned for further talks this semester.