161 research outputs found
Easton supported Jensen coding and projective measure without projective Baire
We prove that it is consistent relative to a Mahlo cardinal that all sets of
reals definable from countable sequences of ordinals are Lebesgue measurable,
but at the same time, there is a set without the Baire property.
To this end, we introduce a notion of stratified forcing and stratified
extension and prove an iteration theorem for these classes of forcings.
Moreover we introduce a variant of Shelah's amalgamation technique that
preserves stratification. The complexity of the set which provides a
counterexample to the Baire property is optimal.Comment: 142 page
Embeddings into outer models
We explore the possibilities for elementary embeddings , where
and are models of ZFC with the same ordinals, , and
has access to large pieces of . We construct commuting systems of such maps
between countable transitive models that are isomorphic to various canonical
linear and partial orders, including the real line
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
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