Changing Cofinalities; Equi-Consistency Results
We formulate a condition which is (strongly) preserved by revised countable support iteration, implies ℵ 1 is satisfied e.g. by of Namba forcing, and any ℵ 1-complete forcing. So we can iterate forcing collapsing ℵ 2 up to some large cardinal.
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940 Saharon Shelah
Proper Forcing
Springer-Verlag Berlin Heidelberg GmbH 1982
Author
Saharon Shelah Institute of Mathematics, The Hebrew University Jerusalem, Israel
AMS Subject Classifications (1980): 03E05, 03E35, 03E45, 03E50
ISBN 978-3-540-11593-9
ISBN 978-3-662-21543-2 (eBook)
DOI 10.1007/978-3-662-21543-2 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 "Vervvertungsgesellschaft Wort", Munich.
© by Springer-Verlag Berlin Heidelberg 1982 Originally published by Springer-Verlag Berlin Heidelberg New York in 1982 2146/3140-543210
INTRODUCTION These notes can be viewed and used in several different ways, each has some justification, a collection of papers, a research monograph or a text book. The author has lectured variants of several of the chapters several times: in University of California, Berkeley, 1978, Ch. III , N, V in Ohio State University in Columbus, Ohio 1979, Ch. I,ll and in the Hebrew University 1979/80 Ch. I, II, III, V, and parts of VI. Moreover Azriel Levi, who has a much better name than the author in such matters, made notes from the lectures in the Hebrew University, rewrote them, and they ·are Chapters I, II and part of III , and were somewhat corrected and expanded by D. Drai, R. Grossberg and the author. Also most of XI §1-5 were lectured on and written up by Shai Ben David. Also our presentation is quite self-contained. We adopted an approach I heard from Baumgartner and may have been used by others: not proving that forcing work, rather take axiomatically that it does and go ahead to applying it. As a result we assume only knowledge of naive set theory (except some iso-
lated points later on in the book). The idea of this approach is that otherwise when the student learns what is axiomatic set theory and how you can show by forcing that CH may fail ( and that CH holds by learning something on L) the Course is finished. But he has only a vague idea of the rich possibilities in forcing, and no idea how to use them. I think the direct approach is more appealing. Also some other parts were written and rewritten Ch. N, was written from notes of Rens from the lecture in Berkeley suffered a heavy criticism of a referee, and thus was rewritten,
then Ron Holtzman corrected it and
expanded it. In Chapter IX, Uri Avraham found various errors, it appeared in [Sh Bid] and was revised later. Chapter X appeared in [Sh 81] and was revised
later.
IV
On the other hand most material was accumulated and not rewritten after the author's knowledge expanded (this is the true reason why some central theorems are not immediately proved in the general form, each Chapter is in fact a paper, though sometimes with references to previous ones. This may serve a
