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Forcing and Independence Proofs

Cooordinated Project, January 2025

Coordination: Dr. Yurii Khomskii

Participants:

Simone Testino
Jonathan Osser
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 ⊩	Kunen 2011: Chapter IV, pp. 244 - 252 (except Lemma IV.2.15) Paul Cohen: The Discover of Forcing Extra material, for fun.
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)	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.

Zoom Link for online meetings: https://uva-live.zoom.us/j/86475080221

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

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