Synopsis for C1.2b: Axiomatic Set Theory

Level: M-level Method of Assessment: Written examination.
Weight: Half-unit (OSS paper code 2B61).

Recommended Prerequisites

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.

Overview

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.

Synopsis

A review of the axioms of ZF set theory. The recursion theorem for the set of natural numbers and for the class of ordinals. 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 (). The fact that implies the Axiom of Choice. Some advanced cardinal arithmetic. The fact that implies the Generalized Continuum Hypothesis.

For the review of ZF set theory:

1. D. Goldrei, Classic Set Theory (Chapman and Hall, 1996).

For course topics (and much more):

1. K. Kunen, Set Theory: An Introduction to Independence Proofs (North Holland, 1983) (now in paperback). Review: Chapter 1. Course topics: Chapters 3, 4, 5, 6 (excluding section 5).

Further Reading

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