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 in S17-0405, except from 12 Jun 2023 to 14 Jul 2023.
Schedule
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Fridays, 4:00 PM – 6:00 PM
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S17-0405, NUS Campus
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12 May 2023 – 4 Aug 2023
(No meetings 12 Jun – 14 Jul)
Literature & Sources
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R. Mansfield, G. Weitkamp. Recursive Aspects of Descriptive Set Theory (1985).
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T. Jech. Set Theory (2003).
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K. Kunen. Set Theory (2011).
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J. Barwise. Admissible Sets and Structures: An Approach to Definability Theory (1975).
Topics Covered
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Consequences of $\mathsf{AD}$ — $\aleph_1$ is measurable, every subset of $\mathbb{R}$ is Lebesgue measurable.
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Descriptive Set Theory — $\Pi_1^1$ normal form, Mostowski absoluteness theorem, Shoenfield absoluteness theorem.
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Large Cardinals & Determinacy — $0^\sharp$, elementary embeddings, analytic determinacy.
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Admissible Set Theory — Kripke-Platek set theory, $\omega$-models, well-founded models, admissible ordinals.
Session Log
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.
Descriptive set theory: Projective hierarchy, $\Pi_1^1$ normal form.
Determinacy: Infinite games, open determinacy.
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.
Martin's measure: Recursive trees, Martin's measure, Martin's cone theorem.
$\mathsf{AD}$ and Lebesgue measure.
26 May 2023
Shoenfield absoluteness theorem: $\kappa$-Suslin sets, tree representation of $\mathbf{\Sigma_2^1}$ sets, Shoenfield absoluteness theorem.
2 Jun 2023
$0^\sharp$ and indiscernibles: Silver indiscernibles, $0^\sharp$, basic consequences of $0^\sharp$.
Analytic determinacy: $0^\sharp$ exists $\to$ $\Sigma_1^1$-$\mathsf{AD}$.
Analytic determinacy: $0^\sharp$ exists $\to$ $\Sigma_1^1$-$\mathsf{AD}$.
9 Jun 2023
Kripke-Platek set theory: Axioms of $\mathsf{KP}$, $\omega$-models, models of $\mathsf{KP}$ and their properties.
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.
$\Sigma_1^1$-$\mathsf{AD} \to 0^\sharp$ exists: Basic properties of Friedman's set.
28 Jul 2023
$\Sigma_1^1$-$\mathsf{AD} \to 0^\sharp$ exists: Main proof.
Equivalences of $0^\sharp$: Kunen's theorem.
Equivalences of $0^\sharp$: Kunen's theorem.
4 Aug 2023
My research: Ramsey theory, Ramsey spaces and MAD families.
Note: The materials above are informal and may be updated at any time without notice. Readers should use them with appropriate caution.