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