Models of ZF-Set Theory
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223 Ulrich Feigner Universitat Heidelberg, Heidelberg/Deutschland
Models of ZF-Set Theory
Springer-Verlag Berlin· Heidelberg· NewYork 1971
AMS Subject Classifications (1970): 02K05, 02K 15, 02K 20, 04-02, 04A 25
ISBN 3-540-05591-6 Springer Verlag Berlin' Heidelberg· New York ISBN 0-387-05591-6 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 Number 73-173745. Printed in Germany.
Offsetdruck: Julius Beltz, HemsbachlBergstr.
PREFACE This set of notes is part and parcel of a series of lectures given in 1970 from February up to June at the Department of Mathematics at the State University at Utrecht and they were intended only as an aid to the people attending those lectures. They were written as the course developed. In fact, in aspect, they are no more than prelecture
of their official looking scribblings. In most
cases complete and detailed proofs are given in these notes but in some few places there are only short indications to the proof. This occurred when a result was only slighty touched in order to round up the presentation. It was the aim of these lectures to provide an exposition of some of the basic techniques and results of the theory of models of Set Theory. This theory covers a wide field and it is not possible to compress it into one series of lectures. We have chosen as our theme the construction of Godel's model L of constructible sets, the construction of FraenkelMostowskiSpecker models (containing ungrounded sets like x
= {x})
and Cohengeneric models. As an
introduction (chapter I), the axiomatization of ZermeloFraenkel is given, some basic concepts and Levy's
of reflection.
Chapter II c6ntains Godel's relative cons
proof for the axiom
of choice, the generalized continuum hypothesis and the axiom of constructibility. Chapters III and IV contain the methods of FraenkelMostowskiSpecker and of P.Cohen in a
setting and various
applications. Many of these applications have not yet appeared in print.
As an aid to the reader several informal discussions and
explanations are included. Although we have attempted to reduce mistakes and Obscurities to a minimum,we should be glad to have our attention drawn to any indiscretion the reader may discover in the text. To finish this preface I wish to express my gratitude to Dvvan Dalen and the Department of ]\'Iathel'latics at Utrecht for the kind invitation t.o spend a Y8ar at this Institute.Thanks are due to R.de Vrijerand K.Rasl'lussen for correcting several misprints and to
KeILe r for typi
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