• Introduction
• Relative consistency proofs
• Formal language and meta-language
• Relativization and examples
• Absoluteness and Δ₀-formulas

• Jech: Chapter 12, pp. 161-164
• Kunen: I.16 until p. 69
• To refresh some memory, Kunen I.15 and Jech p 155-161 are useful

• Classes and schemas
• Tarski's theorem (truth predicate not def.)
• The formal relation ⊨ vs. relativization
• Examples of Δ₀-formulas

• Jech: p 162 ff.
• For an introduction on relativization, absoluteness etc., the old (1980) edition of Kunen is a bit clearer than the new. See Chapter IV (p 110) from "Kunen 1980 Edition"

• ZFC-axioms relativized
• Models of fragments of ZFC
• V_λ ⊨ ZFC \ Replacement for all limits > ω
• H_κ ⊨ ZFC \ Power Set for all regular cardinals > ω
• V_ω ⊨ ZFC \ Infinity
• Strongly inaccessible cardinals
• Revision of the Löwenheim-Skolem theorem for first-order logic

• Jech, p. 167 (for inaccessibles)
• 1980 Kunen: p. 113 - 117 (for relativization of axioms)
• 1980 Kunen: p. 130 - 133 (for H_κ and strongly inaccessibles)

• Cofinality, regular and singular cardinals
• Reflection theorems
• Mostowski collapse
• Introduction to L

• Jech p. 31 - 33 (cofinality)
• Jech p. 168 - 170 (reflection)
• Jech p. 68 - 69 (Mostowski collapse)
• Kunen II.5 (p 129 - 134)

• Definition of L (Gödel's Constructible Universe)
• L ⊨ ZF
• L ⊨ AC
• Absoluteness of the L_α's
• L ⊨ (V = L)
• Minimality of L and related results
• Condensation Lemma and GCH
(Last 3 topics will be repeated next week)

• Kunen p. 135 - 141
• Kunen p. 124 - 125 (absoluteness of L_α)
• Kunen p. 93 - 94 (definition of Def(A))

• Absoluteness of the L_α's (again)
• Minimality of L and related results (again)
• Condensation Lemma and GCH (again)
• Gödel's Incompleteness Theorems (unfinished)

• For L and GCH: see above.
• For Gödel's Incompleteness: TBA.

• Gödel's Incompleteness Theorems
• Martin's Axiom MA

• Kunen: Section III. 3, p. 171-175

• Martin's Axiom MA
• An application of MA in measure theory

• Kunen: Section III. 3, p. 171-175
• Kunen, p. Lemma III.3.28 (p. 179) (In the notation of the theorem, m = least cardinal for which MA fails and add(N) = least cardinal for which the null-ideal is not additive)

• Introduction to forcing
• Generic extensions
• Properties of M[G]
• The semantic forcing relation

• Kunen: Chapter IV, p. 244 - 252 (except Lemma IV.2.15)

• The syntactic forcing relation
• The Truth Lemma and Definability Lemma
• Equivalence of the two forcing relations

• Kunen: Chapter IV, p. 250 - 261 (the relevant parts)

Extra material (for fun):

• Paul Cohen: The Discovery of Forcing
• A talk by Paul Cohen (Vienna, 2006)

• M[G] ⊨ ZFC
• Con(ZFC) → Con(ZFC + V ≠ L)
• Forcing to collapse cardinals
• Adding κ-many new reals by Cohen forcing

• Kunen: Lemma IV.2.26 and Theorem IV.2.27
• Kunen: IV.3 (beginning)

• Adding κ-many new reals by Cohen forcing
• Preservation of cardinals
• Delta-systems
• ccc forcings preserve cardinals
• Con(ZFC + ¬ CH)

• Kunen: p. 263 - 265
• Kunen: p. 166 - 167 (Delta-systems)

• König's Theorem
• Nice names for subsets of ω
• Forcing exact value of continuum
• Preview of other results using forcing

• Kunen: 266 - 267