## Capita Selecta: Set Theory

### 2016/2017; 1st Semester

Institute for Logic, Language & Computation
Universiteit van Amsterdam

Instructor: Prof. Dr. Benedikt Löwe
Teaching assistant: Hugo Nobrega
ECTS: 6

Schedule.

Monday 5 September 2016 Historical introduction: the Continuum Problem. General idea of model constructions by adding new objects and preservation of formulas. Definitional expansions. Transitive models and their relevance. Σ1 and Π1 formulas; extensional classes; relativization; axioms of set theory in submodels; von Neumann hierarchy and axioms of set theory. Homework set #1 (due 15 September 2016) Absoluteness; Δ0 formulas; closure of the class of absolute formulas; list of formulas and functions absolute for transitive models of FST–. Homework set #2 (due 20 September 2016) Literature. Victoria Gitman, Joel David Hamkins, Thomas A. Johnstone, What is the theory ZFC without power set?, Mathematical Logic Quarterly 62:4-5 (2016), pp. 391–406. List of formulas and functions absolute for ZF–: ordinals and rank. Σ1 and Π1 formulas and their absoluteness properties; non-absoluteness of the notion of being a cardinal. Absoluteness of notions defined by transfinite recursion over absolute formulas; defining definability; absoluteness of definability; the constructible hierarchy and basic properties. Homework set #3 (due 27 September 2016) ZFC in L; reflection theorem (without proof); absoluteness of the constructible hierarchy; GCH in L. General methodology of making CH false by going to a bigger model; names as descriptions of elementhood in terms of truth values; basic definitions: incompatibility, chains, antichains, c.c.c., density, genericity; existence of generic filters. Homework set #4 (due 4 October 2016) Names and their interpretation; the generic extension; basic properties of the generic extension, including the minimality of the generic extension M[G] among models of ZFC containing M as subclass and G as element; some of the ZFC axioms in M[G]; an example (forcing with partial functions with finite support to get a surjection). Semantic and syntactic forcing relation; properties; density below p; statement of the Forcing Lemma; proof of the equivalence of semantic and syntactic forcing relation from the Forcing Lemma. Homework set #5 (due 11 October 2016) Proof of the Forcing Theorem. The generic model theorem and its proof. Three applications: (1) proof of the consistency of ZFC+V≠L, (2) collapsing an ordinal to become countable; (3) adding many subsets of ω. Homework set #6 (due 18 October 2016) Preservation of cardinals and regular cardinals. Connection between the chain condition and forcing: θ-c.c. implies that all regular cardinals ≥θ are preserved. The Δ-system lemma; proof of chain conditions for forcing partial orders consisting of functions with finite support. Nice names and upper bounds for the size of the continuum. Further topics (without proofs). Exam