# Unit information: Axiomatic Set Theory in 2022/23

Please note: It is possible that the information shown for future academic years may change due to developments in the relevant academic field. Optional unit availability varies depending on both staffing, student choice and timetabling constraints.

Unit name Axiomatic Set Theory MATHM1300 20 M/7 Teaching Block 1 (weeks 1 - 12) Professor. Welch Not open MATH30100 Logic and MATH32000 Set Theory None None School of Mathematics Faculty of Science

## Unit Information

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 further develops 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.

## Your learning on this unit

Learning Objectives

After taking this unit, students should:

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

Transferable Skills

Assimilation and use of novel and abstract ideas.

## How you will learn

The unit will be taught through a combination of

• Face-to-face lectures
• Independent activities such as problem sheets and/or other exercises

## How you will be assessed

100% Timed, open-book 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.

## Resources

If this unit has a Resource List, you will normally find a link to it in the Blackboard area for the unit. Sometimes there will be a separate link for each weekly topic.

If you are unable to access a list through Blackboard, you can also find it via the Resource Lists homepage. Search for the list by the unit name or code (e.g. MATHM1300).

How much time the unit requires
Each credit equates to 10 hours of total student input. For example a 20 credit unit will take you 200 hours of study to complete. Your total learning time is made up of contact time, directed learning tasks, independent learning and assessment activity.