Content:

• Stationary sets;
• Forcing revisited;
• Forcing Axioms: $\mathbf{MA}$;
• Proper forcing; semi-proper forcing; stationary set preserved forcing;
• $\mathbf{PFA}$, $\mathbf{SPFA}$, $\mathbf{MM}$.

Content:

• Restate $\mathbf{PFA}$, $\mathbf{SPFA}$, $\mathbf{MM}$ as well as $\mathbf{PFA}^{++}$, $\mathbf{SPFA}^{++}$, $\mathbf{MM}^{++}$;
• A few words on iterated forcing
• Supercompact Cardinals, Laver functions;
• Forcing $\mathbf{SPFA}^{(++)}$
• Weak reflection principle;
• $\mathbf{MM}\Rightarrow2^{\aleph_1} = \aleph_2$.

Notice: This is a Reading note of the Handbook article written by Zeman & Schindler, about basics of fine structure theory. The notation may vary but the enumeration of theorems, lemmas and corollaries are the same.

Things I have written so far

• Part I:
  • Rudimentarily Closed & Definability;
  • $S^A_\gamma$ and $<^A_\gamma$;
• Part II:
  • $\Sigma_1$-definability of $S^A_\gamma$ and $<^A_\gamma$;
  • $\Sigma_1$-Satisfaction and the Skolem function;
  • Condensation Lemma;
• Part III:
  • Acceptability & Consequences;
  • $\Sigma_1$-Projectum.
• Part IV:
  • Reducts & Good Parameters;
  • Downward & Upward Extension.

This is a collective course note taken in Prof. Simon Thomas’ course Axiomatic Set Theory of Rutgers University, which is held on Spring semester, 2019. The main topic of this course is forcing, forcing axioms such as $\mathbf{MA}$, Open Coloring Axiom, Axiom A of Baumgartner and the Proper Forcing Axiom of Shelah. Also, this course discussed the relation among themselves and basic independent statements like $\mathbf{CH}$. The main reference would be Kunen’s book and Jech’s book. If there is any mistakes or comments, please feel free to contact me.

《先天性》，节选自阿尔弗雷德·朱尔斯·艾耶尔《语言，真理和逻辑》
收录在《数学哲学》第三编：数学真理

我感觉这篇文章的主要思想还是和我很贴近的。主要在以下方面：

1. 经验论的数学哲学：不存在关于现实的先天知识。数学无关乎现实世界，数学命题是分析命题。在现实世界的（“出乎意料的”）有效性是纯粹偶然的。对于更一般的科学来说，所有科学命题都是“大概的假设”。
2. 数学是有意义的，但数学命题的意义并非在于其在物理世界的映照，而是在于论证分析命题在我们的世界（一个低智商人群组成的世界）是合理的。“一个具有无限智力的人对逻辑和数学毫无兴趣。”
3. 批驳康德主义唯理论：“5+7=12”不是综合命题，是分析命题，错误的根源在于康德使用了非逻辑的论据；几何学是分析的且无关乎人类空间直觉的，因为几何学也不描述物理世界：康德受到了时代的限制。（康德的这个错误常常使得现代数学家忍俊不禁。）

Content:

• Finish the last theorem of the last lecture: Force by a stationary set preserving forcing:
  $$(M;\in,I)\xrightarrow[\text{of length } \omega_1]{\text{generic iteration}}(H_{\omega_2}^V;\in,\mathbf{NS}_{\omega_1}^V),$$

  where $M$ is a generically iterable countable transitive structure.

• $\Bbb P_{\max}$ forcing and analysis of $L(\Bbb R)^{\Bbb P_{\max}}$;

• $(\ast)$ and: $\mathbf{MM}^{++}\implies(\ast)$.

Content:

• Show a characterization of precitiousness;
• $V$ is generically iterable with respect to precitious ideals;
• Discussion of effecitive counterexamples to $\mathbf{CH}$.
• Illustrations of Admissible Club Guessing(ACG)$\implies \mathfrak{u}_2 = \omega_2$.
• Prove ACG follows from $\mathbf{MM}$.

Content:

• discuss some aspects of stationary sets;
• $\mathbf{MM}\implies 2^{\aleph_1}=\aleph_2$;
• effective counterexample to $\mathbf{CH}$.

“理论要用来培养未来指挥官的智力，更准确地说，理论应当促使他们自修，而不是跟着他们一起上战场。这好比高明的老师要做的是启发和促进学生发展智力，而不是一辈子牵着他们走路一样。”

评论：这段话应用范围之广使得《战争论》一书的哲学性显露无疑。我现在怀疑这句话是否蕴含了某种认识论的哲学倾向，但还不好说。
我们常说，“理论指导实践”，指导到底指的是什么？那么在这里，理论就是指南针，但却不是藏宝图。理论和人的自身理性天赋必须得到良好结合，真正的思想火花才能迸发出来。理论给我们指明可能的方向，在一片原野上设立路标，但就其本身而言，并不指涉特定的进路。然而，个人精力的（可悲的）有限性往往使得我们一生只能追寻一条特定的进路，而这样的人就被尊称为某理论家。

Let $\mathbb P = Fn(\omega_2\times\omega,2)$ be the collection of all the finite partial functions from $\omega_2\times \omega$ to $2$. Our strategy is:

• to firstly find a collection of dense sets $D_{\alpha\beta}$, such that a generic filter $G$ can be build upon;
• to secondly prove that any generic filter of $\mathbb P$ preserves cardinals.

Lemma. 1 $D_{\alpha\beta}$ are dense sets, where
$$D_{\alpha\beta} = {p\in\mathbb P\mid \exists n\in\omega(\langle\alpha,n\rangle\in dom(p), \langle\beta,n\rangle\in dom(p),p(\alpha,n)\neq p(\beta,n))}.$$