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Forcing and Independence Proofs

Cooordinated Project, January 2025

Coordination: Dr. Yurii Khomskii

Participants:
  1. Simone Testino
  2. Jonathan Osser
  3. Zhaorui Hu

Project Description

The aim of this project is to study the theory of forcing and independence proofs, including basic principles of models of set theory, absoluteness and reflection theorems, Martin's Axiom (without its consistency proof), the technical aspects of forcing, and up till the original application of forcing which establishes the consistency of ZFC + ¬CH.

The students study the material independently, assisted by several group meetings. There are four assignments to complete and submit. In the last week of January, students give talks presenting a specific segment of the material (not the whole material needs to be covered). Successful evaluation of the project is based on completion of the assignments and presentations.

Textbooks

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

A note about the notation and conventions in Kunen's textbooks.

Topics

This is a detailed list of topics to be covered including reading references (some sections refer to the same material presented by different authors).

Topic     Reading Material Assignments    
1. Models of Set Theory
  • Class models
  • Relativization
  • Absoluteness
  • Kunen 1980: Chapter IV §2, §3 and §5. (p 112 ff)
  • Jech: Chapter 12, pp. 161-164 (same content, more concise)
  • Kunen 2011: I.16 (p 95 - 102) (same content, new edition)
Assignment 1

Submit your assignment here.

2. Reflection and Collapse
  • Mostowski Collapse
  • Reflection Principles
  • Jech p. 68 - 69 (Mostowski collapse)
  • Jech p. 168 - 170 (reflection)
  • Kunen 1980: Chapter IV §7 (more detailed explanation of Reflection)
  • Kunen 2011: II.5 (p. 129 ff) (another detailed explanation of Reflection)
Extra: The Constructible Universe L
  • The main ideas will be presented in an introductory lecture; you can read the corresponding section for further details, but it is not obligatory and there are no assignments on this section.
  • Kunen 2011: II.6, pp. 134 - 141.
3. Martin's Axiom MA
  • Definition of the axiom
  • Basic properties

    • Remark: the axiom may seem arbitrary, but it is introduced here as a way of getting used to the combinatorics of forcing
  • Kunen 2011: Section III.3 until incl. Lemma III.3.15, pp. 171-175.
  • Kunen 2011: Lemma III.2.6, pp. 166 - 167 (Delta-Systems Lemma)
Assignment 2

Submit your assignment here.

4. Introduction to forcing
  • The general idea
  • Generic extensions
  • Properties of M[G]
  • The semantic forcing relation ⊩
5. The technicalities of forcing
  • The syntactic forcing relation ⊩*
  • The Truth Lemma and Definability Lemma
  • Equivalence of the two forcing relations
Assignment 3

Submit your assignment here.

6. The ZFC Axioms
  • M[G] ⊨ ZFC
  • Kunen 2011: Lemma IV.2.15, Lemma IV.2.26 and Theorem IV.2.27
7. Forcing ¬CH.
  • Adding κ-many new reals by Cohen forcing
  • Preservation of cardinals
  • ccc forcings preserve cardinals
  • Con(ZFC + ¬CH)
  • Kunen 2011: pp. 263 - 265
Assignment 4

Submit your assignment here.



Meeting schedule

Date     What Where Notes    
1. Tuesday 7 January (online) Introductory Meeting     online
2. Monday 13 January Mini-lecture on V=L.     L. 1.12 (Lab 42) Notes

Student Presentation Schedule

Date & Time     Who     Topic Room Notes    
1. Friday 14.02, 13:15   Siomne Testino The forcing idea, P-names and generic extensions L 0.11
2. Friday 14.02, 15:15§ Jonathan Osser TBA L 0.11
3. Friday 14.02, 17:15 Zhaorui Hu Consistency of ZFC + not-CH L 0.11