- Go to previous talk.
(At Randall’s request, this entry will be more detailed than usual.)
Remark 1. is club in , so any is stationary as a subset of iff it is stationary as a subset of . It follows that proper forcing preserves stationary subsets of .
Remark 2. Proper forcing extensions satisfy the countable covering property with respect to , namely, if is proper, then any countable set of ordinals in is contained in a countable set of ordinals in . We won’t prove this for now, but the argument for Axiom A posets resembles the general case reasonably:
Given a name for a countable set of ordinals in the extension, find an appropriate regular and consider a countable elementary containing , , and any other relevant parameters. One can then produce a sequence such that
- Each is in .
- , where enumerates the dense subsets of in .
Let for all . Then , so is a countable set of ordinals in containing in . A density argument completes the proof.
Woodin calls a poset weakly proper if the countable covering property holds between and . 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 to without adding bounded subsets of ) or Namba forcing (changing the cofinality of to while preserving are not weakly proper, and account for some of the usefulness of over .
The following is obvious:
Fact. Assume is weakly proper. Then either adds no new -sequences of ordinals, or else it adds a real.
The relation between the reals and the -sequences of ordinals in the presence of strong forcing axioms like is a common theme I am exploring through these talks.