## Set theory seminar -Forcing axioms and inner models II

In this second talk I proved the equivalence of Bagaria’s, and Goldstern-Shelah’s formulation of the bounded forcing axiom for a poset ${\mathbb P}$ that preserves $\omega_1$.

We presented several characterizations of club subsets of ${\mathcal P}_{\omega_1}(X)$ for $X$ uncountable. We then defined when a forcing notion is proper and provided some basic examples of proper forcings, namely ccc, $\sigma$-closed forcings, and their products.

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