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.

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