I hosted a reading seminar in the Summer 2023 at NUS. This reading seminar focused on descriptive set theory, determinacy, and various results in set theory that use recursion theoretic methods to prove. We met every Friday 4-6pm from 12 May 2023 to 4 Aug 2023 at S17-0405, except for the weeks of 12 Jun 2023 to 14 Jul 2023.
The topics we covered include:
- Consequences of $\mathsf{AD}$ - $\aleph_1$ is measurable, every subset of $\mathbb{R}$ is Lebesgue measurable.
- Descriptive Set Theory - $\Pi_1^1$ normal form, Mostowski absoluteness theorem, Shoenfield absoluteness theorem.
- Large cardinals and determinacy - $0^\sharp$, elementary embeddings, analytic determinacy.
- Admissible set theory - Kripke-Platek set theory, $\omega$-models, well-founded models, admissible ordinals.
The sources which I used include:
- Recursive Aspects of Descriptive Set Theory, Richard Mansfield and Galen Weitkamp (1985).
- Set Theory, Thomas Jech (2003).
- Set Theory, Kenneth Kunen (2011).
- Admissible Sets and Structures: An Approach to Definability Theory, Jon Barwise (1975).
The notes are informal and they may change anytime without update. Readers should use them at their own risk.
| # | Date | Topics | Notes |
|---|---|---|---|
| 1 | 12 May 2023 | Background: Large cardinals, measurable cardinals, constructible universe. Descriptive set theory: Projective hierarchy, $\Pi_1^1$ normal form. Determinacy: Infinite games, open determinacy. | Link |
| 2 | 19 May 2023 | Determinacy: $\mathsf{AD}$ and its relationship with $\mathsf{AC}$. Martin’s measure: Recursive trees, Martin’s measure, Martin’s cone theorem. $\mathsf{AD}$ and Lebesgue measure. | Link |
| 3 | 26 May 2023 | Shoenfield absoluteness theorem: $\kappa$-Suslin sets, tree representation of $\mathbf{\Sigma_2^1}$ sets, Shoenfield absoluteness theorem. | Link |
| 4 | 2 Jun 2023 | $0^\sharp$ and indiscernibles: Silver indiscernibles, $0^\sharp$, basic consequences of $0^\sharp$. Analytic determinacy: $0^\sharp$ exists $\to$ $\Sigma_1^1$-$\mathsf{AD}$. | Link |
| 5 | 9 Jun 2023 | Kripke-Platek set theory: Axioms of $\mathsf{KP}$, $\omega$-models, models of $\mathsf{KP}$ and their properties. | Link Addedum |
| 6 | 21 Jul 2023 | Kripke-Platek set theory: Well-founded models. $\Sigma_1^1$-$\mathsf{AD} \to 0^\sharp$ exists: Basic properties of Friedman’s set. | Link |
| 7 | 28 Jul 2023 | $\Sigma_1^1$-$\mathsf{AD} \to 0^\sharp$ exists: Main proof. Equivalences of $0^\sharp$: Kunen’s theorem. | Link |
| 8 | 4 Aug 2023 | My research: Ramsey theory, Ramsey spaces and MAD families. |