## Selected Topics in Set Theory: Constructibility

### Coordination: Dr. Yurii Khomskii

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

### Constructibility

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:
• Models of set theory and absoluteness
• Reflection Theorems
• Basic properties of the constructible universe L
• AC in L
• GCH in L
• Potentially additional topics, e.g.: diamond and combinatorial principles, Suslin trees, definable well-order of the reals, regularity properties for projective sets. <\li>

#### Textbook

We will use the following textbooks:
• Kenneth Kunen, Set Theory (2011 edition) PDF.
• Kenneth Kunen, An Introduction to Independence Proofs (1980) PDF (an older version of the same textbook but better in some respects).
• Thomas Jech, Set Theory (2000 edition). PDF

### Presentations

 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). F1.15 (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) F1.15 (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) F1.15 (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) F1.15 (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) F1.15 (ILLC Seminar room)