Proper Forcing
1st Edition
Published on 8. October 1982
Book
Paperback/Softback
XXXII, 500 pages
978-3-540-11593-9 (ISBN)
Description
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 Univer sity 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).
More details
Series
Language
English
Place of publication
Heidelberg
Germany
Publishing group
Springer Berlin
Target group
Professional and scholarly
Research
Product notice
Paperback (trade)
Dimensions
Height: 23.5 cm
Width: 15.5 cm
Thickness: 27 mm
Weight
1630 gr
ISBN-13
978-3-540-11593-9 (9783540115939)
DOI
10.1007/978-3-662-21543-2
Schweitzer Classification
Other editions
New editions
Saharon Shelah
Proper and Improper Forcing
Book
12/1997
2nd Edition
Springer
€175.48
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Content
Introducing forcing.- The consistency of CH (the continuum hypothesis).- On the consistency of the failure of CH.- More on the cardinality and cohen reals.- Equivalence of forcings notions, and canonical names.- Random reals, collapsing cardinals and diamonds.- The composition of two forcing notions.- Iterated forcing.- Martin Axiom and few applications.- The uniformization property.- Maximal almost disjoint families of subset of ?.- Introducing properness.- More on properness.- Preservation of properness under countable support iteration.- Martin Axiom revisited.- On Aronszajn trees.- Maybe there is no ?2-Aronszajn tree.- Closed unbounded subsets of ?1 can run away from many sets.- On oracle chain conditions.- The omitting type theorem.- Iterations of -c.c. forcings.- Reduction of the main theorem to the main lemma.- Proof of main lemma 4.6.- Iteration of forcing notions which does not add reals.- Generalizations of properness.- ?-properness and (E,?)-properness revisited.- Preservation of ?- properness + the ??- property.- What forcing can we iterate without addding reals.- Specializing an Aronszajn tree without adding reals.- Iteration of orcing notions.- A general preservation theorem.- Three known properties.- The PP(P-point) property.- There may be no P-point.- There may exist a unique Ramsey ultrafilter.- On the ?2-chain condition.- The axioms.- Applications of axiom II.- Application of axiom I.- A counterexample connected to preservation.- Mixed iteration.- Chain conditions revisited.- The axioms revisited.- More on forcing not adding ?-sequences and on the diagonal argument.- Free limits.- Preservation by free limit.- Aronszajn trees: various ways to specialize.- Independence results.- Iterated forcing with RCS (revised countable support).- Proper forcing revisited.- Pseudo-completeness.- Specific forcings.- Chain conditions and Avraham's problem.- Reflection properties of S 02: Refining Avraham's problem and precipitous ideals.- Strong preservation and semi-properness.- Friedman's problem.- The theorems.- The condition.- The preservation properties guaranteed by the S-condition.- Forcing notions satisfying the S-condition.- Finite composition.- Preservation of the I-condition by iteration.- Further independence results.- 0 Introduction.- When is Namba forcing semi-proper, Chang Conjecture and games.- Games and properness.- Amalgamating the S-condition with properness.- The strong covering lemma: Definition and implications.- Proof of strong covering lemmas.- A counterexample.- When adding a real cannot destroy CH.- Bound on for ?? singular.- Concluding remarks and questions.- Unif-strong negation of the weak diamond.- On the power of Ext and Whitehead problem.- Weak diamond for ?2 assuming CH.