Course Title (A2) Elementary Set Theory

[For second-year philosophy undergraduates and above, who are specially interested in formal logic, the philosophy of language and logic, the philosophy of science; (credits 3)]

The course is designed for philosophy undergraduates specially interested in formal logic, philosophy of mathematics and philosophy of language and logic. The aim of this course is to provide a nodding acquaintance with elementary properties of sets and basic axioms of set theory. The study will be confined to the well-established Zermelo -Fraenkel set theory. (A course on Elementary Logic is usually assumed, though not so necessary)

Suggested textbooks

H.B. Enderton, Elements of Set Theory, New York: Academic Press, 1977.
K. Kunen, Set Theory - An Introduction to Independence Proofs, Amsterdam: North-Holland, 1980.


Content
Preliminary

.1 Some na鴳e assumptions of set theory
.2 Two aspects of set theory and the development of set theory: A historical survey
.3 The mathematical and philosophical significance of set theory
.4 A first-order language suitable for set theory

1 Basic Axioms for Sets

2 Functions

3 Relations

4. Natural Numbers

4.1 The axiom of infinity and the construction of - the set of natural numbers
4.2 Peano`s postulates
4.3 Recursion theorems on and arithmetic of
4.4 Orderings on
4.5 Peano Systems and Extensions of natural numbers

5 Equinumerous sets and Schr鐰er-Bernstein Theorem

6 Cardinal Numbers

6.1 The need for cardinality
6.2 Finite and infinite sets
6.3 The axiom schema of replacement
7 The Axiom of Choice and Countable Sets

7.1 The axiom of choice
7.2 Countable sets

8 The Arithmetic of Cardinal Numbers

8.1 Cardinal addition
8.2 Cardinal multiplication (product)
8.3 Cardinal exponentiation

9 Zorn`s Lemma, Continuum Hypothesis and Some Variants of the Axiom of Choice

9.1 Zorn`s Lemma
9.2 Continuum Hypothesis
9.3 Some equivalent variants of Zorn`s Lemma

10 Well-orderings

10.1 Well-orderings of sets
10.2 The comparison theorem for well-ordered sets
10.3 Well -ordering Principle

11. Ordinals

11.1 The Notion of Ordinal Numbers
11.2 Some elementary properties of the ordinals
11.3 Ordinal numbers and cardinal numbers

12 Arithmetic of Ordinals

Appendix. Models of ZF- Set Theory

@. 1 Natural models of ZF - set theory
@.2 The class of infinite cardinal number, cofinality and inaccessible cardinals
@.3 Hereditarily-K-sets
@.4 The Axiom of Constructibility