Set Theory: Boolean-Valued Models and Independence Proofs
This monograph is a follow up to the author's classic text Boolean-Valued Models and Independence Proofs in Set Theory, providing an exposition of some of the most important results in set theory obtained in the 20th century--the independence of the continuum hypothesis and the axiom of choice. Aimed at research students and academics in mathematics, mathematical logic, philosophy, and computer science, the text has been extensively updated with expanded introductory material, new chapters, and a new appendix on category theory, and includes recent developments in the field. Numerous exercises, along with the enlarged and entirely updated background material, make this an ideal text for students in logic and set theory.
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2FORCING AND SOME INDEPENDENCE PROOFS
3GROUP ACTIONS ON VB AND THE INDEPENDENCE OF THE AXIOM OF CHOICE
4 GENERIC ULTRAFILTERS AND TRANSITIVE MODELS OF ZFC
5CARDINAL COLLAPSING BOOLEAN ISOMORPHISM AND APPLICATIONS TO THE THEORY OF BOOLEAN ALGEBRAS
6ITERATED BOOLEAN EXTENSIONS MARTINS AXIOM AND SOUSLINS HYPOTHESIS
7 BOOLEANVALUED ANALYSIS
8 INTUITIONISTIC SET THEORY ANDHEYTINGALGEBRAVALUED MODELS
APPENDIX BOOLEAN AND HEYTING ALGEBRAVALUED MODELS AS CATEGORIES
INDEX OF SYMBOLS
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antichain arbitrary arrow assertion assume automorphism axiom of choice bijection Boolean completion Boolean extensions Boolean-valued models called canonical map Cohen collapsing commutes complete Boolean algebra complete Heyting algebra complete subalgebra construction continuum hypothesis Corollary countable define dense subset distributive lattice dom(u dom(v E V(B element equivalent extensionality filter finite first-order following conditions follows immediately formula function functor Hence holds homomorphism implies independence infinite cardinal intuitionistic set theory isomorphism Let G M-generic ultrafilter Maximum Principle model of ZFC nonempty operations ordinal partially ordered set partition of unity Problem Proof prove real numbers recursion relative consistency result S-complete satisﬁes satisfies ccc sentence sequence Set(H Show that V(B Solovay suppose supremum thatM Theorem topology topos transitive model true in V(B ultrafilter uncountable well-founded relation well-ordered whence Zorn’s lemma