# Notes

Here I store some notes. I prepared these notes to help myself understand certain topics, for presentation at seminars, or recording half-baked research ideas. They have not been checked thoroughly and certainly contain typos and errors to taste.

### Descriptive Set Theory

- Perfect set theorems for closed and analytic sets
- Normal forms for \(\Sigma^1_1\), Luzin’s arithmetic example, and prewellorderings
- There is no Borel 2-coloring for the irrational rotation graph
- Mouse capturing phenomena at \(\Delta^1_1\) and \(\Delta^1_2\)
- The largest \(\Pi^1_1\) thin set as a sharp

### Set Theory

- Introduction to Inaccessible Cardinals
- Measurable Cardinals and Elementary Embeddings
- Product Forcing, Iterated Forcing, Continuum Coding, and Forcing the Ground Axiom

### Teaching Materials

- Prerequisites on Large Cardinals (For Fudan Logic Summer School 2022: Large Cardinals beyond Choice)
- Decision tree method for filling out truth tables (teaching material for intro logic)

### Writings in Chinese (中文笔记)

#### 描述集合论 (Descriptive Set Theory)

- 解析集的正则形式与Mostowski绝对性定理 (Normal forms for analytic sets and Mostowski absoluteness theorem)
- Baire space上闭集的树形式, 以及Gale-Stewart定理的证明 (Closed sets on Baire space as trees, and a proof of the Gale-Stewart theorem)
- The Baire space (Logician’s reals)上的拓扑 (Topology on the Baire space, i.e. the Logician’s reals)
- 一些关于选择公理和广义连续统假设的保守性定理 (Conservativity results for AC and GCH)
- 存在非Borel的解析集: Suslin是如何给Lebesgue纠错的 (How Suslin corrected Lebesgue’s mistake)

#### 集合论/一般数学 (Set Theory/General Mathematics)

- Vitali是如何想到不可测集的 (How did Vitali come up with his non-measurable set?)
- 将实数拆分使得份数比实数多 (A partition of the reals into more parts than the reals)
- Wadge博弈的单射与应用 (Invention of Wadge games and their applications)
- 蕴涵不可测集存在的命题 (Statements implying the existence of non-measurable sets)

#### 历史 (History)

- 第一纲集和第二纲集的来源 (Origins of first and second category sets)
- 连续性定义的发展 (Evolution of the definition of continuity)
- 以开集定义拓扑的历史 (Emergence of using open sets in the definition of topology)
- 冯诺依曼对不完备定理的反应 (Von Neumann’s reaction to the incompleteness theorems)

#### 哲学 (Philosophy)

- 哥德尔定理对柏拉图主义的影响 (Gödel’s theorems and their impact on Platonism)
- 集合论的奠基功能 (Foundational roles played by set theory)

#### 教学与科普 (Teaching Materials/For General Public)

- 秩序与混沌：集合论的世纪斗争 part 1, part 2 (Order and Chaos: The Centennial Struggle of Set Theory)
- Magidor的超紧致基数定义，脱殊超幂(Magidor Characterization of Supercompactness, and Generic Ultrapowers)
- 什么是描述集合论 (What is Descriptive Set Theory?)
- 无穷与悖论：数学与语言的边界（科普幻灯片）