Models of Set Theory

1. First order logic and the axioms of set theory 2 1.1. Syntax 2 1.2. Semantics 2 1.3. Completeness, compactness and consistency 3 1.4. Foundations of mathematics and the incompleteness theorems 3 1.5. The axioms 4 2. Review of basic set theory 5 2.1. Classes 5 2.2. Well-founded relations and recursion 5 2.3. Ordinals, cardinals and arithmetic 6 3. The consistency of the Axiom of Foundation 8 4. Elementary Submodels, the Reflection Principle, and the Mostowski Collapse 10 5. Constructibiltiy 13 5.1. Definability 13 5.2. The definition of L and its elementary properties 14 5.3. ZF in L 15 5.4. V = L? 15 5.5. AC and GCH in L 17 6. Forcing 20 6.1. Partial orderings, dense sets and filters 20 6.2. Generic extensions 22 6.3. The forcing relation 24 6.4. ZFC in M [G] 28 7. CH is independent of ZFC 30 7.1. Forcing CH 30 8. Forcing ¬CH 31 8.1. The countable chain condition and preservation of cardinals 32 8.2. Nice names and the size of 2א0 33 9. Martin’s Axiom 35 9.1. Martin’s Axiom and Souslin’s Hypothesis 35 9.2. The Baire Category Theorem and Martin’s Axiom 37 9.3. Iterated forcing 40 9.4. Long iterations 43 References 45