66 research outputs found
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
Laver and set theory
In this commemorative article, the work of Richard Laver is surveyed in its full range and extent.Accepted manuscrip
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
Set-Theoretic Geology
A ground of the universe V is a transitive proper class W subset V, such that
W is a model of ZFC and V is obtained by set forcing over W, so that V = W[G]
for some W-generic filter G subset P in W . The model V satisfies the ground
axiom GA if there are no such W properly contained in V . The model W is a
bedrock of V if W is a ground of V and satisfies the ground axiom. The mantle
of V is the intersection of all grounds of V . The generic mantle of V is the
intersection of all grounds of all set-forcing extensions of V . The generic
HOD, written gHOD, is the intersection of all HODs of all set-forcing
extensions. The generic HOD is always a model of ZFC, and the generic mantle is
always a model of ZF. Every model of ZFC is the mantle and generic mantle of
another model of ZFC. We prove this theorem while also controlling the HOD of
the final model, as well as the generic HOD. Iteratively taking the mantle
penetrates down through the inner mantles to what we call the outer core, what
remains when all outer layers of forcing have been stripped away. Many
fundamental questions remain open.Comment: 44 pages; commentary concerning this article can be made at
http://jdh.hamkins.org/set-theoreticgeology
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