# Ralf Schindler - Talk 1 on Logic Summer School of Fudan University, 2020

Content:

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

# Ralf Schindler - Talk 2 on Logic Summer School of Fudan University, 2020

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

# Draft of My Fine Structure Notes

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.

# Course Notes of Axiomatic Set Theory

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.