An Outline of Algebraic Set Theory with a View Towards Cohen's Model Falsifying the Continuum Hypothesis
Julia Ilin
Abstract:
This thesis is concerned with the area of algebraic set theory. Algebraic set theory was invented by Joyal and Moerdijk [18] with the aim to study set theory from the perspective of category theory. The central notion is a category of classes, given by a triple (E, S, P_s ), consisting of a Heyting pretopos E, a particular class S of arrows of E that are called small maps and an endofunctor P_s : E → E. The small maps provide an abstract notion of smallness on E, whereas the endofunctor P_s should be thought of as generalized powerclass functor. Universes of set theory arise as initial algebras for this functor.
The main goal of this thesis is prove that Cohen’s model negating the continuum hypothesis can be recovered in the algebraic set theory framework. Cohen’s model has already been examined in the filed of topos theory by Tierney. It will be shown that Tierney’s proof translates to the algebraic set theory setting.