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).
Inhaltsverzeichnis
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.
