Axiomatic Set Theory

Gaisi Takeuti, Wilson M. Zaring (auth.)9780387900506, 0387900500, 0387900519

This text deals with three basic techniques for constructing models of Zermelo-Fraenkel set theory: relative constructibility, Cohen’s forcing, and Scott-Solovay’s method of Boolean valued models. Our main concern will be the development of a unified theory that encompasses these techniques in one comprehensive framework. Consequently we will focus on certain funda­ mental and intrinsic relations between these methods of model construction. Extensive applications will not be treated here. This text is a continuation of our book, “I ntroduction to Axiomatic Set Theory,” Springer-Verlag, 1971; indeed the two texts were originally planned as a single volume. The content of this volume is essentially that of a course taught by the first author at the University of Illinois in the spring of 1969. From the first author’s lectures, a first draft was prepared by Klaus Gloede with the assistance of Donald Pelletier and the second author. This draft was then rcvised by the first author assisted by Hisao Tanaka. The introductory material was prepared by the second author who was also responsible for the general style of exposition throughout the text. We have inc1uded in the introductory material al1 the results from Boolean algebra and topology that we need. When notation from our first volume is introduced, it is accompanied with a deflnition, usually in a footnote. Consequently a reader who is familiar with elementary set theory will find this text quite self-contained.

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Table of contents :
Front Matter….Pages i-vii
Introduction….Pages 1-1
Boolean Algebra….Pages 3-24
Generic Sets….Pages 25-34
Boolean σ-Algebras….Pages 35-46
Distributive Laws….Pages 47-50
Partial Order Structures and Topological Spaces….Pages 51-58
Boolean-Valued Structures….Pages 59-63
Relative Constructibility….Pages 64-78
Relative Constructibility and Ramified Languages….Pages 79-86
Boolean-Valued Relative Constructibility….Pages 87-101
Forcing….Pages 102-105
The Independence of V = L and the CH ….Pages 106-113
The Independence of the AC ….Pages 114-120
Boolean-Valued Set Theory….Pages 121-130
Another Interpretation of V (B) ….Pages 131-142
An Elementary Embedding of V [ F 0 ] in V (B) ….Pages 143-147
The Maximum Principle….Pages 148-159
Cardinals in V (B) ….Pages 160-164
Model Theoretic Consequences of the Distributive Laws….Pages 165-168
Independence Results Using the Models V (B) ….Pages 169-174
Weak Distributive Laws….Pages 175-178
A Proof of Marczewski’s Theorem….Pages 179-182
The Completion of a Boolean Algebra….Pages 183-195
Boolean Algebras That Are Not Sets….Pages 196-220
Easton’s Model….Pages 221-226
Back Matter….Pages 227-240