- 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.