Proper Forcing
Published on 11. December 2013
XXXII, 500 pages
978-3-662-21543-2 (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).
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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.
