Introduction to Axiomatic Set Theory

Gaisi Takeuti, Wilson M. Zaring (auth.)9780387053028, 9780387900247, 9783540053026, 0387053026, 0387900241, 3540053026

In 1963, the first author introduced a course in set theory at the University of Illinois whose main objectives were to cover Godel’s work on the con­ sistency of the Axiom of Choice (AC) and the Generalized Continuum Hypothesis (GCH), and Cohen’s work on the independence of the AC and the GCH. Notes taken in 1963 by the second author were taught by him in 1966, revised extensively, and are presented here as an introduction to axiomatic set theory. Texts in set theory frequently develop the subject rapidly moving from key result to key result and suppressing many details. Advocates of the fast development claim at least two advantages. First, key results are high­ lighted, and second, the student who wishes to master the subject is com­ pelled to develop the detail on his own. However, an instructor using a “fast development” text must devote much class time to assisting his students in their efforts to bridge gaps in the text.

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Table of contents :
Front Matter….Pages i-ix
Introduction….Pages 1-3
Language and Logic….Pages 4-6
Equality….Pages 7-9
Classes….Pages 10-14
The Elementary Properties of Classes….Pages 15-22
Functions and Relations….Pages 23-34
Ordinal Numbers….Pages 35-55
Ordinal Arithmetic….Pages 56-72
Relational Closure and the Rank Function….Pages 73-81
The Axiom of Choice and Cardinal Numbers….Pages 82-99
Cofinality, the Generalized Continuum Hypothesis, and Cardinal Arithmetic….Pages 100-110
Models….Pages 111-120
Absoluteness….Pages 121-142
The Fundamental Operations….Pages 143-152
The Gödel Model….Pages 153-184
Silver Machines….Pages 185-198
Applications of Silver Machines….Pages 199-214
Introduction to Forcing….Pages 215-222
Forcing….Pages 223-228
Back Matter….Pages 229-246