# Forcing over arithmetic: a second-order approach

This is the inaugural talk in the CUNY Models of Peano Arithmetic seminar for the Spring 2018 semester, on Wednesday, 7 February 2018.

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 Schmerlâ€™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, $\Pi^1_k\text{-}\mathsf{CA}$.