課程資訊

Elementary Set Theory

107-1

Phl7713

124EM2990

3.0

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)

Content
Preliminary

.1 Some naive 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 Schroder-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

8.2 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 c

The aim of this course is to provide a nodding acquaintance with elementary properties of sets and basic axioms of set theory.

Office Hours

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.

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.

(僅供參考)

 No. 項目 百分比 說明 1. Class performance 20% 2. Weekly assignment 30% 3. End-of-term exam 50%

 課程進度
 週次 日期 單元主題 第1週 9/14 Preliminary 第2週 9/21 Basic Axioms for Sets 第3週 9/28 Functions 第4週 10/05 Relations 第5週 10/12 Natural Numbers 第6週 10/19 Equinumerous sets and Schröder-Bernstein Theorem 第7週 10/26 Cardinal Numbers 第8週 11/02 Cardinal Numbers 第9週 11/09 The Axiom of Choice and Countable Sets 第10週 11/16 The Arithmetic of Cardinal Numbers 第11週 11/23 The Arithmetic of Cardinal Numbers 第12週 11/30 Zorn's Lemma, Continuum Hypothesis and Some Variants of the Axiom of Choice 第13週 12/07 Zorn's Lemma, Continuum Hypothesis and Some Variants of the Axiom of Choice Well-orderings 第14週 12/14 Ordinals 第15週 12/21 Ordinals 第16週 12/28 Arithmetic of Ordinals 第17週 1/04 Arithmetic of Ordinals 第18週 1/11 Final exam