I hosted a reading seminar in Summer 2023 at NUS. This reading seminar focused on descriptive set theory, determinacy, and various results in set theory that are proved using recursion-theoretic methods. We met every Friday, 4-6pm, from 12 May 2023 to 4 Aug 2023 at S17-0405, except for the period from 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 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 may be updated at any time without notice. Readers should use them with appropriate caution.
| # | 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 Addendum |
| 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. |