Advanced Set Theory: Forcing and Independence Proofs

Cooordinated Project, January 2021, ILLC

Coordination: Dr. Yurii Khomskii

  1. Rodrigo Almeida
  2. Anton Chernev
  3. Quentin Gougeon
  4. Lide Madoe Grotenhuis
  5. Steef Hegeman
  6. Søren Brinck Knudstorp
  7. Gian Marco Osso
  8. Daniel Otten
  9. Francesco Ponti
  10. Tibo Rushbrooke
  11. Wouter Smit
  12. Lamia Tawam

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, the constructible sets, Martin's Axiom (without consistency proof), the technical aspects of forcing and a simple application of forcing establishing the consistency of ZFC + ¬CH.

The students will study the material independently, assisted by regular meetings. There will be a few assignments to complete. In the last week of January, students will give talks presenting some segment of the material. Successful evaluation of the project will be based on completion of the assignments and presentations.


We will use the following textbooks:

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


Below is a detailed list of topics to be covered, with reference to the corresponding textbook sections. Relevant assignments will be posted somewhat later.

Topic     Reading Material Assignment    
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)
2. Reflection and Collapse
  • Mostowski Collapse
  • Reflection Principles
  • Jech p. 68 - 69 (Mostowski collapse)
  • Jech p. 168 - 170 (reflection)
  • Kunen 1980: Chapter IV §2 (more detailed explanation of Reflection)
  • Kunen 2011: II.5 (p. 129 ff) (another detailed explanation of Reflection)
Assignment 1

Submit your assignment here.

Extra: The Constructible Universe L
  • The main ideas will be presented in a meeting; you can read the corresonding 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)
4. Introduction to forcing
  • The general idea
  • Generic extensions
  • Properties of M[G]
  • The semantic forcing relation ⊩
Assignment 2

Submit your assignment here.

5. The technicalities of forcing
  • The syntactic forcing relation ⊩*
  • The Truth Lemma and Definability Lemma
  • Equivalence of the two forcing relations
6. The ZFC Axioms
  • M[G] ⊨ ZFC
  • Kunen 2011: Lemma IV.2.15, Lemma IV.2.26 and Theorem IV.2.27
Assignment 3

Submit your assignment here.

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.


Using this Google Doc you can enter your names to plan possible presentations.


All project-related meetings take place here:

Date     What Notes    
1. Wednesday 23 December, 5 pmIntroductory meeting 1

Yurii Khomskii: introduction to consistency proofs, relativisation and reflection principles
Notes from presentation 1
2. Tuesday 5 January, 11 am Meeting 2

Discussion/questions and lecture about L

Notes from presentation 2
3. Tuesday 5 January, 11 am Meeting 3


Notes from discussion
4. Tuesday 19 January, 11 am Meeting 4


Notes from discussion

Note about Replacement in M[G]

5. Tuesday 26 January, 5 pm Meeting 5


Notes from discussion

Student Presentations

Date     Who What Notes    
Wednesday 27 January, 11 am Rodrigo Almeida & Anton Chernev Martin's Axiom Slides
Wednesday 27 January, 1 pm Lamia Tawam Introduction to forcing
Thursday January, 11 am Søren Brinck Knudstorp & Gian Marco Osso The forcing relation and the Forcing Theorem Slides
Thursday 28 January, 3 pm Quentin Gougeon The ZFC axioms in M[G] Slides
Friday 29 January, 11 am Tibo Rushbrooke & Wouter Smit Details on Reflection Theorems Slides
Friday 29 January, 1 pm Lide Madoe Grotenhuis & Daniel Otten Forcing non-CH Slides
Friday 29 January, 3 pm Steef Hegeman The ccc and preservation of cardinals Slides