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

October 1, 2008

(At Randall’s request, this entry will be more detailed than usual.)

Remark 1. $\omega_1$ is club in ${\mathcal P}_{\omega_1}(\omega_1)$, so any $S\subseteq\omega_1$ is stationary as a subset of $\omega_1$ iff it is stationary as a subset of ${\mathcal P}_{\omega_1}(\omega_1)$. It follows that proper forcing preserves stationary subsets of $\omega_1$.

Remark 2. Proper forcing extensions satisfy the countable covering property with respect to $V$, namely, if ${\mathbb P}$ is proper, then any countable set of ordinals in $V^{\mathbb P}$ is contained in a countable set of ordinals in $V$. We won’t prove this for now, but the argument for Axiom A posets resembles the general case reasonably:

Given a name $\dot X$ for a countable set of ordinals in the extension, find an appropriate regular $\theta$ and consider a countable elementary $N\prec H_\theta$ containing $\dot X$, ${\mathbb P}$, and any other relevant parameters. One can then produce a sequence $(p_n)_{n\in\omega}$ such that

1.  Each $p_i$ is in $N$.
2. $p_{i+1}\le_i p_i$.
3. $p_i\in D_i$, where $(D_n)_{n\in\omega}$ enumerates the dense subsets of ${\mathbb P}$ in $N$.

Let $p\le_i p_i$ for all $i$. Then $p\Vdash \dot X\subseteq N$, so $N\cap{\sf ORD}$ is a countable set of ordinals in $V$ containing $X$ in $V^{\mathbb P}$. A density argument completes the proof.

Woodin calls a poset ${\mathbb P}$ weakly proper if the countable covering property holds between $V$ and $V^{\mathbb P}$. Not every weakly proper forcing is proper, but the countable covering property is a very useful consequence of properness. On the other hand, standard examples of stationary set preserving posets that are not proper, like Prikry forcing (changing the cofinality of a measurable cardinal $\kappa$ to $\omega$ without adding bounded subsets of $\kappa$) or Namba forcing (changing the cofinality of $\omega_2$ to $\omega$ while preserving $\omega_1$ are not weakly proper, and account for some of the usefulness of ${\sf MM}$ over ${\sf PFA}$.

The following is obvious:

Fact. Assume ${\mathbb P}$ is weakly proper. Then either ${\mathbb P}$ adds no new $\omega$-sequences of ordinals, or else it adds a real.

The relation between the reals and the $\omega$-sequences of ordinals in the presence of strong forcing axioms like ${\sf PFA}$ is a common theme I am exploring through these talks.

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.