Forcing and Independence Proofs

Cooordinated Project, January 2024

Coordination: Dr. Yurii Khomskii

Participants:
  1. Qian Chen
  2. Spyros Dialiatsis
  3. Fatima Scha
  4. Tenyo Takahashi
  5. Orestis Tsakakos

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 will study the material independently, assisted by several group meetings. There will be a four assignments to complete and submit. In the last week of January, the students give talks presenting a segment of the material. Successful evaluation of the project is based on completion of the assignments and presentations.

Textbooks

We will use the following textbooks:

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

Topics

Below is a detailed list of topics to be covered, with reference to the corresponding textbook sections.

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.



Preliminary Meetings

Date     What Notes    
1. Wednesday 10 January (online) Discussion about models of set theory, absoluteness and reflection. Mini-lecture on the constructible universe L.     Notes  
1. Wednesday 17 January General discussion and questions.     Notes  


Student Presentations

Date     Who     Topic Room Notes    
1. Wednesday 31 January, 15:00 - 16:30     Qian Chen Reflection Theorems F 1.15   (Seminar Room) Slides
2. Thursday 1 February, 13:00 - 14:30 Orestis Tsakakos Martin's Axiom F 1.15   (Seminar Room)
3. Thursday 1 February, 17:00 - 18:30 Fatima Scha Introduction and main concepts of forcing F 1.15   (Seminar Room)
4. Friday 2 February, 13:30 - 15:00 Spyros Dialiatsis Forcing non-CH (the basic idea) F 3.20  
5. Friday 2 February, 15:00 - 16:30 Tenyo Takahashi Forcing non-CH (ccc and preservation of cardinals) F 3.20