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 $\mathbf{CH}$.

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