# Ralf Schindler - Talk 4 on Logic Summer School of Fudan University, 2020

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 $\mathbf{MM}$.

# Ralf Schindler - Talk 3 on Logic Summer School of Fudan University, 2020

Content:

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

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