21 research outputs found
The Hurewicz dichotomy for generalized Baire spaces
By classical results of Hurewicz, Kechris and Saint-Raymond, an analytic
subset of a Polish space is covered by a subset of if and
only if it does not contain a closed-in- subset homeomorphic to the Baire
space . We consider the analogous statement (which we call
Hurewicz dichotomy) for subsets of the generalized Baire space
for a given uncountable cardinal with
, and show how to force it to be true in a cardinal
and cofinality preserving extension of the ground model. Moreover, we show that
if the Generalized Continuum Hypothesis (GCH) holds, then there is a cardinal
preserving class-forcing extension in which the Hurewicz dichotomy for
subsets of holds at all uncountable regular
cardinals , while strongly unfoldable and supercompact cardinals are
preserved. On the other hand, in the constructible universe L the dichotomy for
sets fails at all uncountable regular cardinals, and the same
happens in any generic extension obtained by adding a Cohen real to a model of
GCH. We also discuss connections with some regularity properties, like the
-perfect set property, the -Miller measurability, and the
-Sacks measurability.Comment: 33 pages, final versio
Strongly uplifting cardinals and the boldface resurrection axioms
We introduce the strongly uplifting cardinals, which are equivalently
characterized, we prove, as the superstrongly unfoldable cardinals and also as
the almost hugely unfoldable cardinals, and we show that their existence is
equiconsistent over ZFC with natural instances of the boldface resurrection
axiom, such as the boldface resurrection axiom for proper forcing.Comment: 24 pages. Commentary concerning this article can be made at
http://jdh.hamkins.org/strongly-uplifting-cardinals-and-boldface-resurrectio
Indestructibility of Vopenka's Principle
We show that Vopenka's Principle and Vopenka cardinals are indestructible
under reverse Easton forcing iterations of increasingly directed-closed partial
orders, without the need for any preparatory forcing. As a consequence, we are
able to prove the relative consistency of these large cardinal axioms with a
variety of statements known to be independent of ZFC, such as the generalised
continuum hypothesis, the existence of a definable well-order of the universe,
and the existence of morasses at many cardinals.Comment: 15 pages, submitted to Israel Journal of Mathematic