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Unit information: Axiomatic Set Theory in 2014/15

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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

Co-requisites

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) M32000.)

School/department School of Mathematics
Faculty Faculty of Science

Description

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.

General Description of the Unit

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.

Further information is available on the School of Mathematics website: http://www.maths.bris.ac.uk/study/undergrad/

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 details

Lectures and exercises to be done by students.

Assessment Details

100% 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

A full text will be handed out.

Alternative & Further Reading:

  • Devlin, K. Constructibility
  • Drake, F. Set Theory
  • Drake, F & Singh, D. Intermediate Set Theory
  • Kunen, K. Set Theory: an Introduction to Independence Proofs

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