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;
• $\mathbf{PFA}$, $\mathbf{SPFA}$, $\mathbf{MM}$.

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 $\mathbf{CH}$.