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Axiomatic Set Theory (MATH M1300)

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Administrative Information

  1. Unit number and title: MATH M1300 Axiomatic Set Theory
  2. Level: M/7
  3. Credit point value: 20 credit points
  4. Year: 07/08
  5. First Given in this form: 2002/03 under the name Constructibility Theory
  6. Unit Organiser: Philip Welch
  7. Lecturer: Prof P. D. Welch
  8. Teaching block: 1
  9. Prerequisites: Essential: Logic MATH30100, Desirable: Set Theory MATH32000. Please discuss with Unit Organiser if you have not taken (or will not be taking) this course.

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 ZFC. 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 3 units Logic and Set Theory.

It is particular;ly pertinent to those interest in, or taking courses in mathematics and philosophy.

Teaching Methods

Lectures and exercises to be done by students.

Learning Objectives

After taking this unit, students should:

  1. Be familiar with the axiomatic basis of the theory of sets.
  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.

Assessment Methods

The assessment mark for Axiomatic Set Theory is calculated from a 2½-hour written examination in April consisting of FIVE questions. A candidate's best FOUR answers will be used for assessment. Calculators are NOT permitted in this examination.

Award of Credit Points

To gain credit points for this unit, students must

  • either gain a pass mark (50 or over) for the unit
  • or gain a mark of at least 30 for the unit, and make satisfactory attempts at the homework assignments.

Transferable Skills

Assimilation and use of novel and abstract ideas.

Texts

 

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

Syllabus

 

The Axioms of Zermelo-Fraenkel Set Theory with Choice

 Class terms, relativisations to modesl; absoluteness

Consistency proofs, reflection theorem 

Closed and unbounded sets, stationary sets;

Regular and singular cardinals, cofinality;  inaccessible cardinals;

 Goedel's Def  function, and the definition of the constructible hierarchy L

The Consistency of AC and GCH with ZFC