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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


Monday 5 September 2016 15-17
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.
Tuesday 6 September 2016 17-19
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)
Thursday 15 September 2016 11-13
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.
Monday 19 September 2016 15-17
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.
Tuesday 20 September 2016 17-19
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)

Monday 26 September 2016 15-17
ZFC in L; reflection theorem (without proof); absoluteness of the constructible hierarchy; GCH in L.
Tuesday 27 September 2016 17-19
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)

Monday 3 October 2016 15-17
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).
Tuesday 4 October 2016 17-19
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)

Monday 10 October 2016 15-17
Proof of the Forcing Theorem.
Tuesday 11 October 2016 17-19
The generic model theorem and its proof. Three applications: (1) proof of the consistency of ZFC+VL, (2) collapsing an ordinal to become countable; (3) adding many subsets of ω.

Homework set #6 (due 18 October 2016)

Monday 17 October 2016 15-17
Preservation of cardinals and regular cardinals. Connection between the chain condition and forcing: θ-c.c. implies that all regular cardinals ≥θ are preserved.
Tuesday 18 October 2016 17-19
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).
Tuesday 25 October 2016 15-17
Last update: 31 October 2016.