36 research outputs found
Laver and set theory
In this commemorative article, the work of Richard Laver is surveyed in its full range and extent.Accepted manuscrip
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
Forcing indestructibility of MAD families
AbstractLet A⊆[ω]ω be a maximal almost disjoint family and assume P is a forcing notion. Say A is P-indestructible if A is still maximal in any P-generic extension. We investigate P-indestructibility for several classical forcing notions P. In particular, we provide a combinatorial characterization of P-indestructibility and, assuming a fragment of MA, we construct maximal almost disjoint families which are P-indestructible yet Q-destructible for several pairs of forcing notions (P,Q). We close with a detailed investigation of iterated Sacks indestructibility
Kurepa-trees and Namba-forcing
We show that compact cardinals and {\rm MM} are sensitive to -closed
forcings for arbitrarily large . This is done by adding 'regressive'
-Kurepa-trees in either case. We argue that the destruction of
regressive Kurepa-trees with {\rm MM} requires the use of Namba forcing
Combinatorial Properties and Dependent choice in symmetric extensions based on L\'{e}vy Collapse
We work with symmetric extensions based on L\'{e}vy Collapse and extend a few
results of Arthur Apter. We prove a conjecture of Ioanna Dimitriou from her
P.h.d. thesis. We also observe that if is a model of ZFC, then
can be preserved in the symmetric extension of in terms of
symmetric system , if
is -distributive and is -complete.
Further we observe that if is a model of ZF + , then
can be preserved in the symmetric extension of in terms of
symmetric system , if
is -strategically closed and is
-complete.Comment: Revised versio
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
Inner models with large cardinal features usually obtained by forcing
We construct a variety of inner models exhibiting features usually obtained
by forcing over universes with large cardinals. For example, if there is a
supercompact cardinal, then there is an inner model with a Laver indestructible
supercompact cardinal. If there is a supercompact cardinal, then there is an
inner model with a supercompact cardinal \kappa for which 2^\kappa=\kappa^+,
another for which 2^\kappa=\kappa^++ and another in which the least strongly
compact cardinal is supercompact. If there is a strongly compact cardinal, then
there is an inner model with a strongly compact cardinal, for which the
measurable cardinals are bounded below it and another inner model W with a
strongly compact cardinal \kappa, such that H_{\kappa^+}^V\subseteq HOD^W.
Similar facts hold for supercompact, measurable and strongly Ramsey cardinals.
If a cardinal is supercompact up to a weakly iterable cardinal, then there is
an inner model of the Proper Forcing Axiom and another inner model with a
supercompact cardinal in which GCH+V=HOD holds. Under the same hypothesis,
there is an inner model with level by level equivalence between strong
compactness and supercompactness, and indeed, another in which there is level
by level inequivalence between strong compactness and supercompactness. If a
cardinal is strongly compact up to a weakly iterable cardinal, then there is an
inner model in which the least measurable cardinal is strongly compact. If
there is a weakly iterable limit \delta of <\delta-supercompact cardinals, then
there is an inner model with a proper class of Laver-indestructible
supercompact cardinals. We describe three general proof methods, which can be
used to prove many similar results