Notes
These range from expository to teaching to research notes. They have not been checked thoroughly and certainly contain typos and errors to taste. Consume at your own risk ;)
Descriptive Set Theory
- Properness of definable forcing need not be projective – Negative consistency result for Question 2.1.4 in Zapletal’s book
- Absolutely \(\Delta^1_2\) sets have constructible generalized Borel codes
- Descriptive complexity measure-preserving transformations with roots. – This mostly relies on a cute trick having to do with non-ergodic MPTs. The ergodic case, which is more interesting, is solved with the help of GPT-5.6. See this note here.
- Perfect set theorems for closed and analytic sets
- Normal forms for \(\Sigma^1_1\), Luzin’s arithmetic example, and prewellorderings
- Mouse capturing phenomena at \(\Delta^1_1\) and \(\Delta^1_2\)
- The largest \(\Pi^1_1\) thin set as a sharp
- Separators of complete coanalytic pairs are complete coanalytic – This is merely a learning exercise for myself, which involves translating known recursion-theoretic proof into more classical descriptive-set-theoretic terms.
Definable Equivalence Relations
- Finite-index extensions of essentially free countable Borel equivalence relations
- A locally countable coanalytic graph not generated by countably many Real-Ordinal definable functions
- Two questions on \(E_0\)-like generic equivalence (2026 Tianyuan Mathematical Logic Problem List 8.1 and 8.3)
- There is no Borel 2-coloring for the irrational rotation graph
- Two questions on infinite-time decidable equivalence relations
Set Theory
- Countable ordinal-definable set in Cohen extension to L only contains ordinal-definable elements, uniformly – (Entirely by GPT-6. Answers this MO question)
- Second-Order Constructibility and HOD Without AC
- Introduction to Inaccessible Cardinals
- Measurable Cardinals and Elementary Embeddings
- Product Forcing, Iterated Forcing, Continuum Coding, and Forcing the Ground Axiom