Proper and improper forcing
The book deals with set-theoretic independence results (independence from the usual set-theoretic ZFC axioms), in particular for problems on the continuum. Consequently, the theory of iterated forcing is developed. The author gives a complete presentation of the theory of proper forcing and its relatives, starting from the beginning and avoiding the metamathematical considerations. In addition to particular consistency results, the author shows methods which can be used for such independence results. Many of these are presented in an "axiomatic" framework (a la Martin's axiom) for this reason. The main aim of the book is thus to enable a researcher interested in an independence result of the approprate kind, to have much of the work done for him, thus allowing him to quote general results.
27 pages matching player I chooses in this book
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Forcing Basic Facts
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adding reals Aronszajn tree assume Axiom belongs cf(a Claim clause clearly closed unbounded subset cofinality Cohen real collapse compatible condition contradiction countable set covering model define by induction Definition dense subset elementary submodel equivalent filter forcing notion function hence holds implies induction hypothesis infinite initial segment iterated forcing Lemma Levy collapse lim(T limit ordinal maximal antichain Ni-complete Note P-name P-point partial order Pi,Qi player I chooses player II wins pn+i pre-dense subset preservation Proof proper forcing RCS iteration regular cardinal Remark satisfies the c.c.c. satisfies the k-c.c semiproper sequence Skolem Hull Souslin stationary set stationary subset strongly inaccessible subset of wi successor ordinal suffices supercompact Suppose Theorem trivial ultrafilter uncountable upper bound w-sequence winning strategy