2 research outputs found
The weakly compact reflection principle need not imply a high order of weak compactness
The weakly compact reflection principle
states that is a weakly compact cardinal and every weakly compact
subset of has a weakly compact proper initial segment. The weakly
compact reflection principle at implies that is an
-weakly compact cardinal. In this article we show that the weakly
compact reflection principle does not imply that is
-weakly compact. Moreover, we show that if the weakly compact
reflection principle holds at then there is a forcing extension
preserving this in which is the least -weakly compact
cardinal. Along the way we generalize the well-known result which states that
if is a regular cardinal then in any forcing extension by
-c.c. forcing the nonstationary ideal equals the ideal generated by the
ground model nonstationary ideal; our generalization states that if is
a weakly compact cardinal then after forcing with a `typical' Easton-support
iteration of length the weakly compact ideal equals the ideal
generated by the ground model weakly compact ideal
Forcing a -like principle to hold at a weakly compact cardinal
Hellsten \cite{MR2026390} proved that when is
-indescribable, the \emph{-club} subsets of provide a
filter base for the -indescribability ideal, and hence can also be
used to give a characterization of -indescribable sets which resembles
the definition of stationarity: a set is
-indescribable if and only if for every
-club . By replacing clubs with -clubs in the
definition of , one obtains a -like principle
, a version of which was first considered by Brickhill and
Welch \cite{BrickhillWelch}. The principle is consistent with
the -indescribability of but inconsistent with the
-indescribability of . By generalizing the standard
forcing to add a -sequence, we show that if is
-weakly compact and holds then there is a
cofinality-preserving forcing extension in which remains
-weakly compact and holds. If is
-indescribable and holds then there is a
cofinality-preserving forcing extension in which is -weakly
compact, holds and every weakly compact subset of has
a weakly compact proper initial segment. As an application, we prove that,
relative to a -indescribable cardinal, it is consistent that
is -weakly compact, every weakly compact subset of has a
weakly compact proper initial segment, and there exist two weakly compact
subsets and of such that there is no for
which both and are weakly compact.Comment: Changed title and added citations to Brickhill-Welc