Mathematical Logic Seminar: How Effective is Forcing?

Abstract:  Since Cohen’s development of forcing to demonstrate the independence of the Continuum Hypothesis from ZFC, the technique has become ubiquitous within set theory. Given how essential forcing is, it is natural to ask questions along the lines of “how constructive is this process?” or “how close is forcing to being computable?”  These questions require a formal interpretation to be meaningful, and we will discuss a couple developed by Hampkins, Miller, and Williams in their 2020 paper “Forcing as a Computational Process.”  We also demonstrate some shortcomings of this approach.