74 research outputs found
Perfect subsets of generalized Baire spaces and long games
We extend Solovay's theorem about definable subsets of the Baire space to the
generalized Baire space , where is an uncountable
cardinal with . In the first main theorem, we show
that that the perfect set property for all subsets of
that are definable from elements of is consistent
relative to the existence of an inaccessible cardinal above . In the
second main theorem, we introduce a Banach-Mazur type game of length
and show that the determinacy of this game, for all subsets of
that are definable from elements of
as winning conditions, is consistent relative to the
existence of an inaccessible cardinal above . We further obtain some
related results about definable functions on and
consequences of resurrection axioms for definable subsets of
Measurable cardinals and good -wellorderings
We study the influence of the existence of large cardinals on the existence
of wellorderings of power sets of infinite cardinals with the property
that the collection of all initial segments of the wellordering is definable by
a -formula with parameter . A short argument shows that the
existence of a measurable cardinal implies that such wellorderings do
not exist at -inaccessible cardinals of cofinality not equal to
and their successors. In contrast, our main result shows that these
wellorderings exist at all other uncountable cardinals in the minimal model
containing a measurable cardinal. In addition, we show that measurability is
the smallest large cardinal property that interferes with the existence of such
wellorderings at uncountable cardinals and we generalize the above result to
the minimal model containing two measurable cardinals.Comment: 14 page
Continuous reducibility and dimension of metric spaces
If is a Polish metric space of dimension , then by Wadge's lemma,
no more than two Borel subsets of can be incomparable with respect to
continuous reducibility. In contrast, our main result shows that for any metric
space of positive dimension, there are uncountably many Borel subsets
of that are pairwise incomparable with respect to continuous
reducibility.
The reducibility that is given by the collection of continuous functions on a
topological space is called the \emph{Wadge quasi-order} for
. We further show that this quasi-order, restricted to the Borel
subsets of a Polish space , is a \emph{well-quasiorder (wqo)} if and
only if has dimension , as an application of the main result.
Moreover, we give further examples of applications of the technique, which is
based on a construction of graph colorings
A hierarchy of Ramsey-like cardinals
We introduce a hierarchy of large cardinals between weakly compact and
measurable cardinals, that is closely related to the Ramsey-like cardinals
introduced by Victoria Gitman, and is based on certain infinite filter games,
however also has a range of equivalent characterizations in terms of elementary
embeddings. The aim of this paper is to locate the Ramsey-like cardinals
studied by Gitman, and other well-known large cardinal notions, in this
hierarchy
Infinite computations with random oracles
We consider the following problem for various infinite time machines. If a
real is computable relative to large set of oracles such as a set of full
measure or just of positive measure, a comeager set, or a nonmeager Borel set,
is it already computable? We show that the answer is independent from ZFC for
ordinal time machines (OTMs) with and without ordinal parameters and give a
positive answer for most other machines. For instance, we consider, infinite
time Turing machines (ITTMs), unresetting and resetting infinite time register
machines (wITRMs, ITRMs), and \alpha-Turing machines for countable admissible
ordinals \alpha
Preserving levels of projective determinacy by tree forcings
We prove that various classical tree forcings -- for instance Sacks forcing,
Mathias forcing, Laver forcing, Miller forcing and Silver forcing -- preserve
the statement that every real has a sharp and hence analytic determinacy. We
then lift this result via methods of inner model theory to obtain
level-by-level preservation of projective determinacy (PD). Assuming PD, we
further prove that projective generic absoluteness holds and no new equivalence
classes classes are added to thin projective transitive relations by these
forcings.Comment: 3 figure
Canonical Truth
We introduce and study a notion of canonical set theoretical truth, which
means truth in a `canonical model', i.e. a transitive class model that is
uniquely characterized by some -formula. We show that this notion of truth
is `informative', i.e. there are statements that hold in all canonical models
but do not follow from ZFC, such as Reitz' ground model axiom or the
nonexistence of measurable cardinals. We also show that ZF++AD
has no canonical models. On the other hand, we show that there are canonical
models for `every real has sharp'. Moreover, we consider `theory-canonical'
statements that only fix a transitive class model of ZFC up to elementary
equivalence and show that it is consistent relative to large cardinals that
there are theory-canonical models with measurable cardinals and that
theory-canonicity is still informative in the sense explained above
The Hurewicz dichotomy for generalized Baire spaces
By classical results of Hurewicz, Kechris and Saint-Raymond, an analytic
subset of a Polish space is covered by a subset of if and
only if it does not contain a closed-in- subset homeomorphic to the Baire
space . We consider the analogous statement (which we call
Hurewicz dichotomy) for subsets of the generalized Baire space
for a given uncountable cardinal with
, and show how to force it to be true in a cardinal
and cofinality preserving extension of the ground model. Moreover, we show that
if the Generalized Continuum Hypothesis (GCH) holds, then there is a cardinal
preserving class-forcing extension in which the Hurewicz dichotomy for
subsets of holds at all uncountable regular
cardinals , while strongly unfoldable and supercompact cardinals are
preserved. On the other hand, in the constructible universe L the dichotomy for
sets fails at all uncountable regular cardinals, and the same
happens in any generic extension obtained by adding a Cohen real to a model of
GCH. We also discuss connections with some regularity properties, like the
-perfect set property, the -Miller measurability, and the
-Sacks measurability.Comment: 33 pages, final versio
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