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.

Set theory seminar -Forcing axioms and inner models

September 12, 2008

Today I started a series of talks on “Forcing axioms and inner models” in the Set Theory Seminar. The goal is to discuss a few results about strong forcing axioms and to see how these axioms impose a certain kind of rigidity’ to the universe.

I motivated forcing axioms as trying to capture the intuition that the universe is wide’ or saturated’ in some sense, the next natural step after the formalization via large cardinal axioms of the intuition that the universe is tall.’

The extensions of ${\sf ZFC}$ obtained via large cardinals and those obtained via forcing axioms share a few common features that seem to indicate their adoption is not arbitrary. They provide us with reflection principles (typically, at the level of the large cardinals themselves, or at small cardinals, respectively), with regularity properties (and determinacy) for many pointclasses of reals, and with generic absoluteness principles.

The specific format I’m concentrating on is of axioms of the form ${\sf FA}({\mathcal K})$ for a class ${\mathcal K}$ of posets, stating that any ${\mathbb P}\in{\mathcal K}$ admits filters meeting any given collection of $\omega_1$ many dense subsets of ${\mathbb P}$. The proper forcing axiom ${\sf PFA}$ is of this kind, with ${\mathcal K}$ being the class of proper posets. The strongest axiom to fall under this setting is Martin’s maximum ${\sf MM}$, that has as ${\mathcal K}$ the class of all posets preserving stationary subsets of $\omega_1$.

Of particular interest is the `bounded’ version of these axioms, which, if posets in ${\mathcal K}$ preserve $\omega_1$, was shown by Bagaria to correspond precisely to an absoluteness statement, namely that $H_{\omega_2}\prec_{\Sigma_1}V^{\mathbb P}$ for any ${\mathbb P}\in{\mathcal K}$.

In the next meeting I will review the notion of properness, and discuss some consequences of ${\sf BPFA}$.