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Set Theory (MATH 32000)

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Administrative Information

  1. Unit number and title: MATH 32000 Set Theory
  2. Level: H/6
  3. Credit point value: 20 credit points
  4. Year: 07/08
  5. First Given in this form: 1996-97
  6. Unit Organiser: Philip Welch
  7. Lecturer: Prof P Welch
  8. Teaching block: 2
  9. Prerequisites: Is a preliminary for the Level M unit Axiomatic Set Theory (MATH M1300).

Unit aims

To introduce the students to the general theory of sets, as a foundational and as an axiomatic theory.

General Description of the Unit

The aim is to make the course of general interest to students who are not planning to specialize in mathematical logic or the Level M Axiomatic Set Theory, but of special interest to those who are.

Set Theory can be regarded as a  foundation for all, or most, of mathematics, in that any mathematical concept  can be formulated as being about sets. The course shows how we can represent the natural numbers as sets and how principles such as proof by mathematical induction can be seen as  being built up from very primitive notions about sets.

We shall see how the pitfalls of the various early "set theoretic paradoxes" such as that of Russell ("the set of all sets that do not contain themselves") were avoided.  We develop Cantor's theory of transfinite ordinal numbers and their arithmetic through the introduction of his most substantial contribution to mathematics: the notion of wellordering.  We shall see
how an "arithmetic of the infinite" can be developed that extends naturally the arithmetic of the finite we all know. We shall introduce the principle of ordinal induction and recursion
along the ordinals to extend that of mathematical induction and recursion along the natural numbers. Cantor's famous proof of the uncountability of the real continuum by a diagonal argument, and his  revolutionary discovery that there were different "orders of infinity" - indeed infinitely many such - will feature prominently in our basic study of infinite cardinal numbers and their arithmetic.

We shall see how axiom sets can be used to develop this theory, and indeed the whole cumulative hierarchy of sets of mathematical discourse.  There will be discussion of the axioms system ZF developed by Zermelo and Fraenkel in the wake of Cantor's work, and about the role the Axiom of Choice plays in set theory.

Texts: Full lecture notes will be provided.  See also: "Elements of Set Theory" by H.B. Enderton, Academic Press QA248 END Queens Library (3 copies) which contains  all the subject matter we shall cover. 

Relation to Other Units

Set Theory may be regarded as the foundation for all mathematics. This course, although not a formal prerequisite, is a preliminary for the level 4 Axiomatic Set Theory M1300.

For students interested in the philosophy of mathematics,: this course is related to a number of units in the philosophy department in philosophy of mathematics.  It should thus be of interest to any joint Maths/Philosoohy degree students. 

Teaching Methods

Lectures and sheets.

Learning Objectives

The student should come away from this course with a basic understanding of such topics as the theory of partial orderings and well orderings, cardinality, ordinal numbers, and the role of the Axiom of Choice. He or she should also have become aware of the role of set theory as a foundation for mathematics, and of the part that axiomatic set theory has to play.

Assessment Methods

The assessment mark for Set Theory is calculated from a 2 ½-hour written examination in May/June consisting consisting of FIVE questions. A candidate's best FOUR answers will be used for assessment. Calculators are NOT permitted be used in this examination.

Award of Credit Points

Credit points are gained by:

  • passing the unit
  • obtaining an examination mark of 30 or over and submiting satisfactory attempts at the problem sheets.

Texts

Lecture notes will be provided.

Syllabus

The basic principles and definitions of set theory.

The axiomatic definition of natural number.

Well-ordering and the Axiom of Choice.

Definition by transfinite induction

Ordinal and cardinal numbers.

The cumulative hierarchy of sets

Axiomatic Set Theory.