# C1.4 Axiomatic Set Theory - Material for the year 2019-2020

This course presupposes basic knowledge of First Order Predicate Calculus up to and including the Soundness and Completeness Theorems, together with a course on basic set theory, including cardinals and ordinals, the Axiom of Choice and the Well Ordering Principle.

16 lectures.

### Assessment type:

- Written Examination

Inner models and consistency proofs lie at the heart of modern Set Theory, historically as well as in terms of importance. In this course we shall introduce the first and most important of inner models, Gödel's constructible universe, and use it to derive some fundamental consistency results.

A review of the axioms of ZF set theory. Absoluteness, the recursion theorem. The Cumulative Hierarchy of sets and the consistency of the Axiom of Foundation as an example of the method of inner models. Levy's Reflection Principle. Gödel's inner model of constructible sets and the consistency of the Axiom of Constructibility ($V=L$). $V=L$ is absolute. The fact that $V=L$ implies the Axiom of Choice. Some advanced cardinal arithmetic. The fact that $V=L$ implies the Generalized Continuum Hypothesis.

For the review of ZF set theory and the prerequisites from Logic:

- D. Goldrei,
*Classic Set Theory*(Chapman and Hall, 1996). - K. Kunen,
*The Foundations of Mathematics*(College Publications, 2009).

For course topics (and much more):

- K. Kunen,
*Set Theory*(College Publications, 2011) Chapters (I and II).

- K. Hrbacek and T. Jech,
*Introduction to Set Theory*(3rd edition, M Dekker, 1999).