Selected Topics in Set Theory: Constructibility

January 2020, ILLC

Coordination: Dr. Yurii Khomskii

  1. Teodor Calinoiu
  2. James Carr
  3. David de Graaf
  4. Jonathan Osinski
  5. Frank Westers


In 1938 Gödel constructed a model of set theory, known as the Constructible Universe L, in which the Axiom of Choice (AC) and the Generalized Continuum Hypothesis (GCH) are satisfied, proving that neither AC nor GCH could be refuted on the basis of ZF. The model L is generally seen as a "minimal model of set theory", and has since been shown to satisfy many other interesting properties. In this project, we cover the basic theory of Gödel's Constructible Universe L and related topics. In particular:


We will use the following textbooks:

A note about the notation in Kunen's textbooks.


Date     Time     Who    What     Pages     Where    
Wednesday 7 Jan     13-15 All Preparatory meeting F1.15

(ILLC Seminar room)

Monday 27 Jan     15-17 Frank Westers
  • Models of Set Theory
  • Relativization
  • Relative consistency
  • Δ0-formulas and absoluteness
  • Jech: Chapter 12, p. 161-164 for a clear overview (you may skip Tarski's theorem).
  • 1980 Kunen: IV.1 and IV.2 (p.110-117) for more detail.
  • The same is contained in the 2011 Kunen p. 95-96 and 107-109 (but the presentation is more confusing in my opinion).

(ILLC Seminar room)

Tuesday 28 Jan     11-13 Teodor Calinoiu
  • Models of fragments of ZFC
  • Vλ ⊨ ZFC \ Replacement for all limits > ω
  • Hκ ⊨ ZFC \ Power Set for all regular cardinals > ω
  • Vω ⊨ ZFC \ Infinity
  • (if possible) Vκ for inaccessible κ
  • Jech, p. 167 (for inaccessibles)
  • 1980 Kunen: p. 113-117 (for relativization of axioms)
  • 1980 Kunen: p. 130-133 (for Hκ and strongly inaccessibles)

(ILLC Seminar room)

Tuesday 28 Jan     14-16 David de Graaf
  • The Mostowski Collapse
  • Reflection Theorems
  • Jech: p. 68-69 (for the Mostowski collapse)
  • Jech: p. 168-170 (for a clear overview of Reflection)
  • 2011 Kunen II.5, p. 129-134 (Reflection, in more detail)

(ILLC Seminar room)

Wednesday 29 Jan     10-12 James Carr
  • Definition of L (Gödel's Constructible Universe)
  • Basic properties of L
  • The ZF-axioms in L
  • The Axiom of Choice in L
  • 2011 Kunen: II.6 until/incl. Theorem II.6.11 (p. 134-137)
  • 2011 Kunen: Def. II.6.18, Def. II.6.19 and Theorem II.6.20 (Axiom of Choice)

(ILLC Seminar room)

Wednesday 29 Jan     12.30 - 14.30 Jonathan Osinski
  • The axiom "V=L"
  • Absoluteness of L
  • Minimality of L
  • Condensation Lemma and the GCH in L
  • 2011 Kunen: Lemma II.6.13 until/incl. Lemma II.6.16 (p. 138)
  • You will also need: 2011 Kunen, p. 123-125 (you may skip the details here)
  • 2011 Kunen: Lemma II.6.22 until/incl. Theorem II.6.24
  • For extra help: Jech Theorem 13.16, Lemma 13.17 and Theorem 13.20 (p. 187-191)

(ILLC Seminar room)