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 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.

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

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 ⊩	Kunen 2011: Chapter IV, pp. 244 - 252 (except Lemma IV.2.15) Paul Cohen: The Discover of Forcing Extra material, for fun.	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	Kunen 2011: Chapter IV, pp. 255 - 262 A talk by Paul Cohen, Vienna, 2006 Extra material, for fun (put your volume up, hard to hear)
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.

Presentations

Meetings

	Date	What	Notes
1.	Wednesday 23 December, 5 pm	Introductory 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 Discussion/questions	Notes from discussion
4.	Tuesday 19 January, 11 am	Meeting 4 Discussion/questions	Notes from discussion Note about Replacement in M[G]
5.	Tuesday 26 January, 5 pm	Meeting 5 Discussion/questions	Notes from discussion

Student Presentations