Proper Forcing [electronic resource] / by Saharon Shelah.

Shelah, Saharon author., Author,
Berlin, Heidelberg : Springer Berlin Heidelberg : Imprint: Springer, 1982.
Lecture Notes in Mathematics, 0075-8434 ; 940
Lecture Notes in Mathematics, 0075-8434 ; 940
1 online resource (XXXII, 500 pages)
Logic, Symbolic and mathematical.
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Mathematical Logic and Foundations. (search)
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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).
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
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.
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