Capita Selecta: Set Theory
2016/2017; 1st Semester
Institute for Logic, Language & ComputationUniversiteit van Amsterdam
Instructor: Prof. Dr. Benedikt
Löwe
Teaching assistant: Hugo Nobrega
ECTS: 6
Schedule.
| Monday 5 September 2016 | 15-17 G3.13 | 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 D1.162 |
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 G2.10 |
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 G3.02 | 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 D1.162 | 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 D1.112 | ZFC in L; reflection theorem (without proof); absoluteness of the constructible hierarchy; GCH in L. |
| Tuesday 27 September 2016 | 17-19 D1.162 | 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 D1.112 | 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 D1.162 |
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 D1.162 | Proof of the Forcing Theorem. |
| Tuesday 11 October 2016 | 17-19 D1.162 |
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) |
| Monday 17 October 2016 | 15-17 D1.162 | 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 D1.162 | 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 A1.04 | Exam |