3 research outputs found
Capturing sets of ordinals by normal ultrapowers
We investigate the extent to which ultrapowers by normal measures on
can be correct about powersets for . We
consider two versions of this questions, the capturing property
and the local capturing property
. holds if there is
an ultrapower by a normal measure on which correctly computes
. is a weakening of
which holds if every subset of is
contained in some ultrapower by a normal measure on . After examining
the basic properties of these two notions, we identify the exact consistency
strength of . Building on results of Cummings,
who determined the exact consistency strength of
, and using a forcing due to Apter and Shelah, we
show that can hold at the least measurable
cardinal.Comment: 20 page
Transferring Compactness
We demonstrate that the technology of Radin forcing can be used to transfer
compactness properties at a weakly inaccessible but not strong limit cardinal
to a strongly inaccessible cardinal.
As an application, relative to the existence of large cardinals, we construct
a model of set theory in which there is a cardinal that is
--stationary for all but not weakly compact. This is in
sharp contrast to the situation in the constructible universe , where
being --stationary is equivalent to being
-indescribable. We also show that it is consistent that there
is a cardinal such that is
-stationary for all and , answering a
question of Sakai.Comment: Corrected some typo