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