Topics on Set Theory, MAST 661/ sec. M / 837
Topics on Set Theory, MAST 661/ sec. M / 837
This will be a reading seminar on the method of forcing and its applications.
In the first part of the course will develop the method of forcing and prove the independence of the Continuum Hypothesis from ZFC, and the independence of the Axiom of Choice from ZF. We will cover several technical notions which are crucial for modern day set theory, including: chain conditions; closure and distributivity conditions; product forcing and mutual genericity; collapse forcing; projections and isomorphism of forcing notions.
Given time, we will continue to more advanced applications of forcing, in particular to descriptive set theory. We will start with proving the consistency of ZF + DC + 'all sets of reals are Lebesgue measurable', assuming the existence of an inaccessible cardinal (the Solovay Model).
The evaluation will be based on seminar presentations (70%) and one written assignment (30%). The precise number of presentations will depend on the number of participants.
Projected outline:
1. (3 weeks) Review of relevant notions from axiomatic set theory.
2. (3 weeks) Developing the forcing method and proving the forcing theorem.
3. (3 weeks) Proving independence of CH from ZFC and the independence of AC from ZF. Constructing various models of ZF where AC fails.
4. (3 weeks) The Solovay model. Given time we will cover more connections between forcing and descriptive set theory.
Semester: Fall 2026 (12 weeks)
Credits: 3 units
Location: Concordia University
Meetings: Mondays & Wednesdays 10:15 - 11:00(+ε), LB 921-4.
[First meeting: Wednesday September 9
Note: there will be no meetings on Sep 14, Oct 12, 19, Nov 16, Dec 9.]
Prereqs: Familiarity with axiomatic set theory, McGill's MATH 488 or equivalent.
Textbook: no textbook is required.
We will use these notes for parts 1 through 3. See also this list of problems. For part 4 we will use Kanamori's 'The Higher Infinite'.
Additional useful resources:
Kunen - Set Theory: An Introduction to Independence Proofs
Jech - Set Theory
Halbeisen - Combinatorial Set Theory: With a Gentle Introduction to Forcing
Andrew Marks' notes on set theory
Steve Jackson's notes on set theory
Spencer Unger's notes on the Solovay model.
You are expected to read the notes thoroughly and go through all the exercises in the notes and the questions in the problem list. Not everything in the notes will be part of a presentation. We should meet to dicuss what to focus on in the presentations. To some extent this is left up to you (which part of the proof to do in detail and which not, etc.).
For Section 2 (review of axiomatic set theory) we should focus (in the presentations) on the following concepts.
Levy heirarchy, absoluteness, V_alpha and H(kappa), Mostowski Collapse.
[Exercise 2.27; Theorem 2.28; Exercise 2.30; Question 3: (2),(3),(4); Theorem 2.31; Theorem 2.63; Exercise 2.64; Question 9 (1)]
Ordinal definability [Section 2.12; Question 8; Question 10]