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Proper Forcing

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...

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Bibliographic Details
Main Author: Shelah, S.
Format: eBook
Language:English
Published: Berlin, Heidelberg Springer Berlin Heidelberg 1982, 1982
Edition:1st ed. 1982
Series:Lecture Notes in Mathematics
Subjects:
Mathematical Logic
Mathematical Logic And Foundations
Online Access:
Collection: Springer Book Archives -2004 - Collection details see MPG.ReNa
Table of Contents:
  • 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.
  • ?-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
  • 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

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