623 research outputs found
Absoluteness via Resurrection
The resurrection axioms are forcing axioms introduced recently by Hamkins and
Johnstone, developing on ideas of Chalons and Velickovi\'c. We introduce a
stronger form of resurrection axioms (the \emph{iterated} resurrection axioms
for a class of forcings and a given
ordinal ), and show that implies generic
absoluteness for the first-order theory of with respect to
forcings in preserving the axiom, where is a
cardinal which depends on ( if is any
among the classes of countably closed, proper, semiproper, stationary set
preserving forcings).
We also prove that the consistency strength of these axioms is below that of
a Mahlo cardinal for most forcing classes, and below that of a stationary limit
of supercompact cardinals for the class of stationary set preserving posets.
Moreover we outline that simultaneous generic absoluteness for
with respect to and for with respect to
with is in principle
possible, and we present several natural models of the Morse Kelley set theory
where this phenomenon occurs (even for all simultaneously). Finally,
we compare the iterated resurrection axioms (and the generic absoluteness
results we can draw from them) with a variety of other forcing axioms, and also
with the generic absoluteness results by Woodin and the second author.Comment: 34 page
Category forcings, , and generic absoluteness for the theory of strong forcing axioms
We introduce a category whose objects are stationary set preserving complete
boolean algebras and whose arrows are complete homomorphisms with a stationary
set preserving quotient. We show that the cut of this category at a rank
initial segment of the universe of height a super compact which is a limit of
super compact cardinals is a stationary set preserving partial order which
forces and collapses its size to become the second uncountable
cardinal. Next we argue that any of the known methods to produce a model of
collapsing a superhuge cardinal to become the second uncountable
cardinal produces a model in which the cutoff of the category of stationary set
preserving forcings at any rank initial segment of the universe of large enough
height is forcing equivalent to a presaturated tower of normal filters. We let
denote this statement and we prove that the theory of
with parameters in is generically invariant
for stationary set preserving forcings that preserve . Finally we
argue that the work of Larson and Asper\'o shows that this is a next to optimal
generalization to the Chang model of Woodin's generic
absoluteness results for the Chang model . It remains open
whether and are equivalent axioms modulo large cardinals
and whether suffices to prove the same generic absoluteness results
for the Chang model .Comment: - to appear on the Journal of the American Mathemtical Societ
Non-Absoluteness of Model Existence at
In [FHK13], the authors considered the question whether model-existence of
-sentences is absolute for transitive models of ZFC, in
the sense that if are transitive models of ZFC with the same
ordinals, and , then if and only if .
From [FHK13] we know that the answer is positive for and under
the negation of CH, the answer is negative for all . Under GCH, and
assuming the consistency of a supercompact cardinal, the answer remains
negative for each , except the case when which is an
open question in [FHK13].
We answer the open question by providing a negative answer under GCH even for
. Our examples are incomplete sentences. In fact, the same
sentences can be used to prove a negative answer under GCH for all
assuming the consistency of a Mahlo cardinal. Thus, the large cardinal
assumption is relaxed from a supercompact in [FHK13] to a Mahlo cardinal.
Finally, we consider the absoluteness question for the
-amalgamation property of -sentences (under
substructure). We prove that assuming GCH, -amalgamation is
non-absolute for . This answers a question from [SS]. The
cases and infinite remain open. As a corollary we get that
it is non-absolute that the amalgamation spectrum of an
-sentence is empty
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