Proper Forcing (Lecture Notes in Mathematics)

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. Saharon Shelah. Bibliography: P. 492-496.