Selected Topics in Set Theory

Selected Topics in Set Theory: Constructibility

January 2020, ILLC

Coordination: Dr. Yurii Khomskii

Participants:

Teodor Calinoiu
James Carr
David de Graaf
Jonathan Osinski
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)
Kenneth Kunen, An Introduction to Independence Proofs (1980) (an older version of the same textbook but better in some respects).
Thomas Jech, Set Theory (2000 edition).

A note about the notation in Kunen's textbooks.

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 V_λ ⊨ ZFC \ Replacement for all limits > ω H_κ ⊨ ZFC \ Power Set for all regular cardinals > ω V_ω ⊨ ZFC \ Infinity (if possible) V_κ for inaccessible κ	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)	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)