Monday 5 September 2016
 1517 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
 1719 D1.162

Σ_{1} and Π_{1} 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
 1113 G2.10

Absoluteness; Δ_{0} 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
62:45 (2016), pp. 391–406.

Monday 19 September 2016
 1517 G3.02
 List of formulas and functions absolute for ZF–: ordinals and rank.
Σ_{1} and Π_{1} formulas and their absoluteness properties; nonabsoluteness
of the notion of being a cardinal.

Tuesday 20 September 2016
 1719 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
 1517 D1.112
 ZFC in L;
reflection theorem (without proof);
absoluteness of the constructible hierarchy; GCH in L.

Tuesday 27 September 2016
 1719 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
 1517 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
 1719 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
 1517 D1.162

Proof of the Forcing Theorem.

Tuesday 11 October 2016
 1719 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
 1517 D1.162

Preservation of cardinals and regular cardinals. Connection between the
chain condition and forcing: θc.c. implies that all regular
cardinals ≥θ are preserved.

Tuesday 18 October 2016
 1719 D1.162

The Δsystem 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
 1517 A1.04
 Exam
