Mykyta Narusevych (PhD student, MFF UK) will give a survey talk regarding forcing in arithmetics. There are no regular talks this semester, so Mykyta’s talk is specific for next week.

Monday 17.3.2025
Time: 16:30, department of logic, C116, Celetna ulice

Title
Forcing in Bounded Arithmetic

Abstract
To show that a theory does not prove a sentence, it is enough to construct a model of the said theory where the sentence fails. While simple in principle, it is pretty hard to implement the above procedure if one is interested in reasonably strong theories, such as theories of arithmetic or sets.

A significant breakthrough in constructing various interesting models came with the famous result of Paul Cohen resolving the Continuum Hypothesis. The forcing method developed by Cohen was subsequently adapted and extended to various other fields and, in particular, to bounded arithmetic.

In this talk, I will present the work of Atserias and Müller, which unified several previous forcing results in bounded arithmetic, including the theorems of Paris and Wilkie, Riis, and Ajtai. I will discuss a partially definable forcing relation and, if time permits, several extensions to the mentioned work.