| 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
|
