367 research outputs found
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
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
Laver and set theory
In this commemorative article, the work of Richard Laver is surveyed in its full range and extent.Accepted manuscrip
Narrow coverings of omega-product spaces
Results of Sierpinski and others have shown that certain finite-dimensional
product sets can be written as unions of subsets, each of which is "narrow" in
a corresponding direction; that is, each line in that direction intersects the
subset in a small set. For example, if the set (omega \times omega) is
partitioned into two pieces along the diagonal, then one piece meets every
horizontal line in a finite set, and the other piece meets each vertical line
in a finite set. Such partitions or coverings can exist only when the sets
forming the product are of limited size.
This paper considers such coverings for products of infinitely many sets
(usually a product of omega copies of the same cardinal kappa). In this case, a
covering of the product by narrow sets, one for each coordinate direction, will
exist no matter how large the factor sets are. But if one restricts the sets
used in the covering (for instance, requiring them to be Borel in a product
topology), then the existence of narrow coverings is related to a number of
large cardinal properties: partition cardinals, the free subset problem,
nonregular ultrafilters, and so on.
One result given here is a relative consistency proof for a hypothesis used
by S. Mrowka to construct a counterexample in the dimension theory of metric
spaces
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