# Axiomatic Set Theory

## Unit aims

To develop the theory of Gödel's universe of constructible sets; to use this model to prove the consistency of various statements of mathematics with the currently accepted axioms of set theory.

## Unit description

It is known that various straightforward mathematical statements are neither provable nor disprovable in the best available axiomatic system of set theory that we have. This system, Zermelo-Fraenkel set theory ("ZF"), provides a theoretical underpinning of all of mathematics, in that any mathematical statement, if provable, can be proven in this system. However certain straightforward statements, e.g., the Axiom of Choice (in one form: "every set can be wellordered") can be neither proved nor disproved in ZF. Another is the Continuum Hypothesis ("CH": that every uncountable set of real numbers can be put in (1-1) correspondence with the set of all real numbers). The course will contain a discussion of the nature of axiomatic systems, the nature of concepts such as "provability", "unprovability" in such systems, and the status of Gödel's famous Incompleteness Theorems (roughly that any axiom system T extending that of, eg, Peano's system for arithmetic cannot prove a statement Con(T) encapsulating the consistency of that formal system) in the setting of set theory.

There will follow an introduction to the axiomatics of ZF together with the construction of "L", a universe of sets invented by Gödel, This allowed him to show that both AC and CH were not disprovable.

If time permits we shall sketch Cohen's 1963 forcing method that showed how the CH was not provable from ZF; or else we may discuss further strong axioms of infinity, or "large cardinals".

## Relation to other units

This is the only unit which develops further the concepts in the Level 6 units Logic and Set Theory.

It is particularly pertinent to those interested in, or taking courses in mathematics and philosophy.

## Learning objectives

After taking this unit, students should:

- Be familiar with the axiomatic basis of the theory of the universe of sets of mathematical discourse.
- Be able to understand the notion of an "inner model" of set theory.
- Be able to understand how such models enable consistency statements.
- Have a working knowledge of the constructibility hierarchy.

## Transferable skills

Assimilation and use of novel and abstract ideas.

## Syllabus

The Axioms of Zermelo-Fraenkel Set Theory with Choice

Class terms, relativisations to models;

Absoluteness and reflection theorems

Introduction to Consistency proofs,

Closed and unbounded sets, stationary sets;

Regular and singular cardinals, cofinality; inaccessible cardinals;

Godel Def function, and the definition of the constructible hierarchy L

The Consistency of AC and GCH with ZFC

## Reading and References

A full text will be handed out.

Alternative & Further Reading

Devlin, K. *Constructibility (also for further reading)*

Drake, F. *Set Theory*

Drake, F & Singh, D. *Intermediate Set Theory*

Further reading: Kunen, K. *Set Theory: an Introduction to Independence Proofs*

**Unit code:** MATHM1300

**Level of study:** M/7

**Credit points:** 20

**Teaching block (weeks):** 1 (1-12)

**Lecturer:** Professor Philip Welch

## Pre-requisites

Essential Prerequisites: MATH30100 Logic; and as a Prerequisite or Co-requisite: Set Theory MATH32000. (However please discuss with Unit Organiser if you have not taken (or will not be taking) 32000.)

## Co-requisites

None

## Methods of teaching

Lectures and exercises to be done by students.

## Assessment methods

The assessment mark for Axiomatic Set Theory is calculated from a 2½-hour written examination. Candidates should answer ALL questions for assessment. Calculators are NOT permitted in this examination.

Raw scores on the examinations will be determined according to the marking scheme written on the examination paper. The marking scheme, indicating the maximum score per question, is a guide to the relative weighting of the questions. Raw scores are moderated as described in the Undergraduate Handbook.