Advanced Set Theory, Winter Semester 2018

Winter Semester 2018


Weekly Content

Lecture    Date     Material     Reading material Photos of blackboard Homework     Practice exercises
1 15 October   
  • 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
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 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
Lecture 2 Photos Homework 2 Practice 2
3 29 October   
  • ZFC-axioms relativized
  • Models of fragments of ZFC
  • Vλ ⊨ ZFC \ Replacement for all limits > ω
  • Hκ ⊨ ZFC \ Power Set for all regular cardinals > ω
  • Vω ⊨ 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 Hκ 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 Lα'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 Lα)
  • Kunen p. 93 - 94 (definition of Def(A))
Lecture 5 Photos Homework 5
6 27 November   
  • Absoluteness of the Lα'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. 171-175
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. 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)
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 κ-many new reals by Cohen forcing
  • Kunen: Lemma IV.2.26 and Theorem IV.2.27
  • Kunen: IV.3 (beginning)
Lecture 11 Photos Homework 11