Spring 2020 Project on Forcing

March-June 2020, ILLC

Coordination: Dr. Yurii Khomskii

Participants:

Anna Dmitrieva
David de Graaf
Jonathan Osinski
Robert Schütz

Project Description

The aim of this project is to study the basic theory of forcing, including basic principles of models of set theory, absoluteness and reflection theorems, also covering 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 (online) meetings. There will be a few assignments to complete. The student participation will be evaluated based on completion of the assignments as well as general participation.

Textbook

We will use the following textbooks:

Kenneth Kunen, Set Theory (2011 edition)
Kenneth Kunen, An Introduction to Independence Proofs (1980) (an older version of the same textbook but better in some respects).
Thomas Jech, Set Theory (2000 edition).

A note about the notation in Kunen's textbooks.

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

Topic Reading Material Homework Assignment

1. Models of Set Theory

Class models

Relativization

Relative Consistency

Absoluteness

Reflection Theorems

Mostowski Collapse

Jech: Chapter 12, pp. 161-164

Kunen 2011: I.16 until p. 69

For an introduction on relativization, absoluteness etc., the old (1980) edition of Kunen is a bit clearer than the new. See Chapter IV (p 110) from "Kunen 1980 Edition"

1980 Kunen: p. 113 - 117 (relativization of axioms)

Jech p. 168 - 170 (reflection)

Jech p. 68 - 69 (Mostowski collapse)

Kunen 2011: Section II.5, p. 129 - 134) (reflection)

Assignment 1
2. Martin's Axiom MA

Definition of the axiom

Basic properties

Remark: the axiom may seem very arbitrary, but it is introduced first as a way of getting used to the terminology used in forcing theory later

(Optional) An interesting application of MA to Lebesgue Measures

Kunen 2011: Section III. 3, p. 171-175 (incl. proof of Lemma III.3.15)

Kunen 2011: Lemma III.2.6, p. 166 - 167 (Delta-Systems Lemma)

3. Introduction to forcing

The general idea

Generic extensions

Properties of M[G]

The semantic forcing relation ⊩

Kunen 2011: Chapter IV, p. 244 - 252 (except Lemma IV.2.15)

Extra material (for fun): Paul Cohen: The Discover of Forcing
Assignment 2

4. The technicalities of forcing

The syntactic forcing relation ⊩^*

The Truth Lemma and Definability Lemma

Equivalence of the two forcing relations

Kunen 2011: Chapter IV, p. 255 - 262

Extra material (for fun): A talk by Paul Cohen, Vienna, 2006 (put your volume up, hard to hear)

5. The ZFC Axioms

M[G] ⊨ ZFC

Con(ZFC) → Con(ZFC + V ≠ L)
Only if you know about L

Kunen 2011: Lemma IV.2.15, Lemma IV.2.26 and Theorem IV.2.27

6. Forcing ¬CH.

Forcing to collapse cardinals

Adding κ-many new reals by Cohen forcing

Preservation of cardinals

Delta-systems

ccc forcings preserve cardinals

Con(ZFC + ¬CH)

Kunen 2011: p. 263 - 265

Assignment 3

7. Finer properties of forcing ¬CH

Nice names for subsets of ω

Forcing exact value of continuum

Kunen 2011: 266 - 267

Assignment 4

Topic	Reading Material	Homework Assignment

1. Models of Set Theory Class models Relativization Relative Consistency Absoluteness Reflection Theorems Mostowski Collapse	Jech: Chapter 12, pp. 161-164 Kunen 2011: I.16 until p. 69 For an introduction on relativization, absoluteness etc., the old (1980) edition of Kunen is a bit clearer than the new. See Chapter IV (p 110) from "Kunen 1980 Edition" 1980 Kunen: p. 113 - 117 (relativization of axioms) Jech p. 168 - 170 (reflection) Jech p. 68 - 69 (Mostowski collapse) Kunen 2011: Section II.5, p. 129 - 134) (reflection)	Assignment 1
2. Martin's Axiom MA Definition of the axiom Basic properties Remark: the axiom may seem very arbitrary, but it is introduced first as a way of getting used to the terminology used in forcing theory later (Optional) An interesting application of MA to Lebesgue Measures	Kunen 2011: Section III. 3, p. 171-175 (incl. proof of Lemma III.3.15) Kunen 2011: Lemma III.2.6, p. 166 - 167 (Delta-Systems Lemma)
3. Introduction to forcing The general idea Generic extensions Properties of M[G] The semantic forcing relation ⊩	Kunen 2011: Chapter IV, p. 244 - 252 (except Lemma IV.2.15) Extra material (for fun): Paul Cohen: The Discover of Forcing	Assignment 2
4. The technicalities of forcing The syntactic forcing relation ⊩^* The Truth Lemma and Definability Lemma Equivalence of the two forcing relations	Kunen 2011: Chapter IV, p. 255 - 262 Extra material (for fun): A talk by Paul Cohen, Vienna, 2006 (put your volume up, hard to hear)
5. The ZFC Axioms M[G] ⊨ ZFC Con(ZFC) → Con(ZFC + V ≠ L) Only if you know about L	Kunen 2011: Lemma IV.2.15, Lemma IV.2.26 and Theorem IV.2.27
6. Forcing ¬CH. Forcing to collapse cardinals Adding κ-many new reals by Cohen forcing Preservation of cardinals Delta-systems ccc forcings preserve cardinals Con(ZFC + ¬CH)	Kunen 2011: p. 263 - 265	Assignment 3
7. Finer properties of forcing ¬CH Nice names for subsets of ω Forcing exact value of continuum	Kunen 2011: 266 - 267	Assignment 4