177 research outputs found
Subcompact cardinals, squares, and stationary reflection
We generalise Jensen's result on the incompatibility of subcompactness with
square. We show that alpha^+-subcompactness of some cardinal less than or equal
to alpha precludes square_alpha, but also that square may be forced to hold
everywhere where this obstruction is not present. The forcing also preserves
other strong large cardinals. Similar results are also given for stationary
reflection, with a corresponding strengthening of the large cardinal assumption
involved. Finally, we refine the analysis by considering Schimmerling's
hierarchy of weak squares, showing which cases are precluded by
alpha^+-subcompactness, and again we demonstrate the optimality of our results
by forcing the strongest possible squares under these restrictions to hold.Comment: 18 pages. Corrections and improvements from referee's report mad
Category forcings, , and generic absoluteness for the theory of strong forcing axioms
We introduce a category whose objects are stationary set preserving complete
boolean algebras and whose arrows are complete homomorphisms with a stationary
set preserving quotient. We show that the cut of this category at a rank
initial segment of the universe of height a super compact which is a limit of
super compact cardinals is a stationary set preserving partial order which
forces and collapses its size to become the second uncountable
cardinal. Next we argue that any of the known methods to produce a model of
collapsing a superhuge cardinal to become the second uncountable
cardinal produces a model in which the cutoff of the category of stationary set
preserving forcings at any rank initial segment of the universe of large enough
height is forcing equivalent to a presaturated tower of normal filters. We let
denote this statement and we prove that the theory of
with parameters in is generically invariant
for stationary set preserving forcings that preserve . Finally we
argue that the work of Larson and Asper\'o shows that this is a next to optimal
generalization to the Chang model of Woodin's generic
absoluteness results for the Chang model . It remains open
whether and are equivalent axioms modulo large cardinals
and whether suffices to prove the same generic absoluteness results
for the Chang model .Comment: - to appear on the Journal of the American Mathemtical Societ
Woodin for strong compactness cardinals
We give the definition of Woodin for strong compactness cardinals, the
Woodinised version of strong compactness, and we prove an analogue of Magidor's
identity crisis theorem for the first strongly compact cardinal.Comment: 20 pages, fixed proof of Theorem 4.1, minor corrections and addition
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