12 research outputs found
Indestructibility of compact spaces
In this article we investigate which compact spaces remain compact under
countably closed forcing. We prove that, assuming the Continuum Hypothesis, the
natural generalizations to -sequences of the selection principle and
topological game versions of the Rothberger property are not equivalent, even
for compact spaces. We also show that Tall and Usuba's "-Borel
Conjecture" is equiconsistent with the existence of an inaccessible cardinal.Comment: 18 page
Weak Indestructibility and Reflection
This work is a part of my upcoming thesis [7]. We establish an
equiconsistency between (1) weak indestructibility for all -degrees
of strength for cardinals in the presence of a proper class of strong
cardinals, and (2) a proper class of cardinals that are strong reflecting
strongs. We in fact get weak indestructibility for degrees of strength far
beyond , well beyond the next inaccessible limit of measurables (of
the ground model). One direction is proven using forcing and the other using
core model techniques from inner model theory. Additionally, connections
between weak indestructibility and the reflection properties associated with
Woodin cardinals are discussed.Comment: 28 page
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
Superstrong and other large cardinals are never Laver indestructible
Superstrong cardinals are never Laver indestructible. Similarly, almost huge
cardinals, huge cardinals, superhuge cardinals, rank-into-rank cardinals,
extendible cardinals, 1-extendible cardinals, 0-extendible cardinals, weakly
superstrong cardinals, uplifting cardinals, pseudo-uplifting cardinals,
superstrongly unfoldable cardinals, \Sigma_n-reflecting cardinals,
\Sigma_n-correct cardinals and \Sigma_n-extendible cardinals (all for n>2) are
never Laver indestructible. In fact, all these large cardinal properties are
superdestructible: if \kappa\ exhibits any of them, with corresponding target
\theta, then in any forcing extension arising from nontrivial strategically
<\kappa-closed forcing Q in V_\theta, the cardinal \kappa\ will exhibit none of
the large cardinal properties with target \theta\ or larger.Comment: 19 pages. Commentary concerning this article can be made at
http://jdh.hamkins.org/superstrong-never-indestructible. Minor changes in v