271 research outputs found
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
The Proper Forcing Axiom and the Singular Cardinal Hypothesis
We show that the Proper Forcing Axiom implies the Singular Cardinal
Hypothesis. The proof is by interpolation and uses the Mapping Reflection
Principle.Comment: 10 page
Generic absoluteness and boolean names for elements of a Polish space
It is common knowledge in the set theory community that there exists a
duality relating the commutative -algebras with the family of -names
for complex numbers in a boolean valued model for set theory . Several
aspects of this correlation have been considered in works of the late 's
and early 's, for example by Takeuti, and by Jech. Generalizing Jech's
results, we extend this duality so as to be able to describe the family of
boolean names for elements of any given Polish space (such as the complex
numbers) in a boolean valued model for set theory as a space
consisting of functions whose domain is the Stone space of , and
whose range is contained in modulo a meager set. We also outline how this
duality can be combined with generic absoluteness results in order to analyze,
by means of forcing arguments, the theory of .Comment: 27 page
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
Absolute model companionship, forcibility, and the continuum problem
Absolute model companionship (AMC) is a strict strengthening of model
companionship defined as follows: For a theory ,
denotes the logical consequences of which are boolean combinations of
universal sentences. is the AMC of if it is model complete and
. The -theory
of algebraically closed field is the model companion of the
theory of but not its AMC as is in
.
We use AMC to study the continuum problem and to gauge the expressive power
of forcing. We show that (a definable version of) is
the unique solution to the continuum problem which can be in the AMC of a
"partial Morleyization" of the -theory "there are class
many supercompact cardinals". We also show that (assuming large cardinals)
forcibility overlaps with the apparently weaker notion of consistency for any
mathematical problem expressible as a -sentence of a (very large
fragment of) third order arithmetic (, the Suslin hypothesis, the
Whitehead conjecture for free groups are a small sample of such problems
).
Partial Morleyizations can be described as follows: let
be the set of first order -formulae; for
, is the expansion of adding
atomic relation symbols for all formulae in and
is the -theory asserting that each -formula
is logically equivalent to the corresponding atomic
formula . For a -theory is the
partial Morleyization of induced by .Comment: This paper systematizes and improves the results appearing in arxiv
submissions arXiv:2101.07573, arXiv:2003.07114, arXiv:2003.0712
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