Unit information: Axiomatic Set Theory in 2020/21

Unit name	Axiomatic Set Theory
Unit code	MATHM1300
Credit points	20
Level of study	M/7
Teaching block(s)	Teaching Block 1 (weeks 1 - 12)
Unit director	Professor. Welch
Open unit status	Not open
Pre-requisites	MATH30100 Logic and MATH32000 Set Theory
Co-requisites	None
School/department	School of Mathematics
Faculty	Faculty of Science

Description including Unit Aims

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.

Intended Learning Outcomes

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.

Teaching Information

The unit will be taught through a combination of

• synchronous online and, if subsequently possible, face-to-face lectures
• asynchronous online materials, including narrated presentations and worked examples
• guided asynchronous independent activities such as problem sheets and/or other exercises
• synchronous weekly group problem/example classes, workshops and/or tutorials
• synchronous weekly group tutorials
• synchronous weekly office hours

Assessment Information

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.

Reading and References

Recommended

• Keith J. Devlin, Constructibility, Springer-Verlag, 1984
• Frank R. Drake, Set Theory: An Introduction to Large Cardinals, North-Holland, 1974
• F.R. Drake and D. Singh, Intermediate Set Theory, Wiley, 1996

Further

• Kenneth Kunen, Set Theory: An Introduction to Independence Proofs, North-Holland, 1980