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;
• $\mathbf{MM}\Rightarrow2^{\aleph_1} = \aleph_2$.

Content:

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

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.

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