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Advanced Set Theory: Forcing and Independence Proofs
Cooordinated Project, January 2021, ILLC
Coordination: Dr. Yurii Khomskii
Participants:- Rodrigo Almeida
- Anton Chernev
- Quentin Gougeon
- Lide Madoe Grotenhuis
- Steef Hegeman
- Søren Brinck Knudstorp
- Gian Marco Osso
- Daniel Otten
- Francesco Ponti
- Tibo Rushbrooke
- Wouter Smit
- 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.
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
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
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2. Reflection and Collapse
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Assignment 1
Submit your assignment here. |
Extra: The Constructible Universe L
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3. Martin's Axiom MA
Remark: the axiom may seem arbitrary, but it is introduced here as a way of getting used to the combinatorics of forcing |
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4. Introduction to forcing
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Assignment 2
Submit your assignment here. |
5. The technicalities of forcing
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6. The ZFC Axioms
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Assignment 3
Submit your assignment here. |
7. Forcing ¬CH.
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Assignment 4
Submit your assignment here. |
Presentations
Using this Google Doc you can enter your names to plan possible presentations.
Meetings
All project-related meetings take place here: https://uva-live.zoom.us/j/86951814480
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 |
5. | Tuesday 26 January, 5 pm | Meeting 5 Discussion/questions | 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 |