We show that, like singular cardinals, and weakly compact cardinals, Jensen's core model K for measures of order zero 4] calculates correctly the successors of JJ onsson cardinals, assuming O Sword does not exist. Namely, if is a JJ onsson cardinal then + = +K , provided that there is no non-trivial elementary embedding j : K ?! K. There are a number of related results in ZF C concerning P() in...

If κ is any strongly unfoldable cardinal, then this is preserved in a forcing extension in which 3κ(reg) fails. This result continues the progression of the corresponding results for weakly compact cardinals, due to Woodin, and for indescribable cardinals, due to Hauser.

If κ is any strongly unfoldable cardinal, then this is preserved in a forcing extension in which 3κ(reg) fails. This result continues the progression of the corresponding results for weakly compact cardinals, due to Woodin, and for indescribable cardinals, due to Hauser.

In this note we characterize the compact weighted Frobenius-Perron operator $p$ on $L^1(Sigma)$ and determine their spectra. We also show that every weakly compact weighted Frobenius-Perron operator on $L^1(Sigma)$ is compact.

We present a characterization of weakly compact cardinals in terms of generalized stationarity. We apply this characterization to construct a model with no partial square sequences. One of the striking features of weak compactness in the large cardinal hierarchy is the wide variety of characterizations of the concept. These characterizations meet many disparate areas of set theory, including me...

We show that Shelah’s Eventual Categoricity Conjecture follows from the existence of class many strongly compact cardinals. This is the first time the consistency of this conjecture has been proven. We do so by showing that every AEC with LS(K) below a strongly compact cardinal κ is < κ tame and applying the categoricity transfer of Grossberg and VanDieren [GV06a]. These techniques also apply t...

Three central combinatorial properties in set theory are the tree property, the approachability property and stationary reflection. We prove the mutual independence of these properties by showing that any of their eight Boolean combinations can be forced to hold at κ, assuming that κ = κ and there is a weakly compact cardinal above κ. If in addition κ is supercompact then we can force κ to be א...

We consider the cardinal invariant CG(X) of the minimal number of weakly compact subsets which generate a Banach space X. We study the behavior of this index when passing to subspaces, its relation with the Lindelöf number in the weak topology and other related questions. A Banach space is weakly compactly generated if there is a weakly compact subset which is linearly dense and weakly Lindelöf...

We study the preservation of the property of L(R) being a Solovay model under proper projective forcing extensions. We show that every ∼ 1 3 strongly-proper forcing notion preserves this property. This yields that the consistency strength of the absoluteness of L(R) under ∼ 1 3 strongly-proper forcing notions is that of the existence of an inaccessible cardinal. Further, the absoluteness of L(R...