Lectures in Set Theory with Particular Emphasis on the Method of Forcing
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217 Thomas J. Jech State University of New York at Buffalo, New York, NYfUSA University of California, Los Angeles, CAfUSA
Lectures in Set Theory with Particular Emphasis on the Method of Forcing
Springer-Verlag Berlin· Heidelberg· New York 1971
AMS Subject Classifications (1970): 02K05, 02 K 15,02 K20, 02K25, 02K99, 04-02
ISBN 3-540-05564-9 Springer-Verlag Berlin' Heidelberg· New York ISBN 0-387-05564-9 Springer-Verlag New York· Heidelberg· Berlin This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher.
© by Springer-Verlag Berlin' Heidelberg 1971. Library of Congress Catalog Card Number70-171872.Printed in Germany. Offsetdruck: Julius Beltz, Hemsbach/Bergstr.
PREFACE The notes contain material covered in the graduate course I gave at the state University of New York in Buffalo in 1969-70. is put on the method of forcing.
As the title suggests, the emphasis
I felt it necessary to provide the notes with the
subtitle because two important parts of set theory are completely ignored here:
the
theory of large cardinals and the descriptive set theory. Many theorems in these notes are relatively new results.
I tried to give everyone credit for the results.
Throughout the text
I may have not been completely
successful since some methods have become a part of "mathematical folklore" and are disseminated mostly by oral communication. who is responsible for forcing.
my
My
special thanks are due to P.
interest in set theory and who taught me the finesses of
Also, I am indebted to
my
former colleagues from Prague, whose enthusiasm
made the work in set theory a real adventure. The notes contain enough material for a two-semester graduate course.
I did
not include any exercises, but I hope that an eager student will find enough enjoyment in working out the details of some of the proofs.
Los Angeles, March 1971
CONTENTS
..
1-
Formulas and classes •
2.
Axioms of Zermelo-Fraenkel
2
3.
Ordinal numbers •
6
1
4. Cardinal numbers
9
5. Finite sets
12
6. Real numbers
13
.
15
Cardinal arithmetic
16
9· Axiom of regularity
19
7· Axiom of choice • 8.
10. Transitive models
20
11.
Constructible sets
32
12.
Consistency of AC
and
35
GCH
13. More on transitive models
36
14.
41
Ordinal definability
15. Ultrapowers
43
16.
Remarks on complete Boolean algebras •
47
17. Method of forcing and Boolean-valued models
51
18.
62
Independence of the continuum hypothesis and collapsing of cardinals • •
19. Two applications of Boolean-valued models in the theory of Boolean algebras
74
20.
Lebesgue measurability
78
21.
Suslin's problem
90
22 .
Mart in I s axio
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