Lecture 
Date 
Material 
Reading material 
Photos of blackboard 
Homework 
Practice exercises 
     
1 
15 October

 Introduction
 Relative consistency proofs
 Formal language and metalanguage
 Relativization and examples
 Absoluteness and Δ_{0}formulas

 Jech: Chapter 12, pp. 161164
 Kunen: I.16 until p. 69
 To refresh some memory, Kunen I.15 and Jech p 155161 are useful
 Lecture 1 Photos 
Homework 1 

2 
22 October

 Classes and schemas
 Tarski's theorem (truth predicate not def.)
 The formal relation ⊨ vs. relativization
 Examples of Δ_{0}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
 Lecture 2 Photos 
Homework 2 
Practice 2 
3 
29 October

 ZFCaxioms 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 LöwenheimSkolem theorem for firstorder 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)
 Lecture 3 Photos 
Homework 3 
Practice 3 
4 
5 November

 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)
 Lecture 4 Photos 
Homework 4 
Practice 4 
5 
12 November

 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))
 Lecture 5 Photos 
Homework 5 

6 
27 November

 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.
 Lecture 6 Photos 
Homework 6 
Practice 6 
7 
3 December

 Gödel's Incompleteness Theorems
 Martin's Axiom MA

 Kunen: Section III. 3, p. 171175
 Lecture 7 Photos 
Homework 7 
Practice 7 
8 
10 December

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

 Kunen: Section III. 3, p. 171175
 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 nullideal is not additive)
 Lecture 8 Photos 
Homework 8 

9 
17 December

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

 Kunen: Chapter IV, p. 244  252 (except Lemma IV.2.15)
 Lecture 9 Photos 
Homework 9 
Practice 9 
10 
7 January

 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):
 Lecture 10 Photos 
Homework 10 
Practice 10 
11 
14 January

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

 Kunen: Lemma IV.2.26 and Theorem IV.2.27
 Kunen: IV.3 (beginning)
 Lecture 11 Photos 
Homework 11 
