This is a talk in CUNY’s MoPA seminar on 14 September 2016.

In previous semesters of this seminar, I have talked about how the technique of forcing, originally developed by Cohen for building models of set theory, can be used to produce models of arithmetic with various properties. In this talk, I will present a forcing proof of Harrington’s theorem on the conservativity of $\mathsf{WKL}_0$ over $\mathsf{RCA}_0$. More formally, any countable model of $\mathsf{RCA}_0$ can be extended to a model of $\mathsf{WKL}_0$ with the same first-order part. As an immediate corollary, we get that any $\Pi^1_1$ sentence provable by $\mathsf{WKL}_0$ is already provable by $\mathsf{RCA}_0$.