20 research outputs found
Menger's covering property and groupwise density
We establish a surprising connection between Menger's classical covering
property and Blass's modern combinatorial notion of groupwise density. This
connection implies a short proof of the groupwise density bound on the
additivity number for Menger's property.Comment: Small update
Template iterations with non-definable ccc forcing notions
We present a version with non-definable forcing notions of Shelah's theory of
iterated forcing along a template. Our main result, as an application, is that,
if is a measurable cardinal and are
uncountable regular cardinals, then there is a ccc poset forcing
. Another
application is to get models with large continuum where the groupwise-density
number assumes an arbitrary regular value.Comment: To appear in the Annals of Pure and Applied Logic, 45 pages, 2
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Set Theory
This meeting covered all important aspects of modern Set Theory, including large cardinal theory, combinatorial set theory, descriptive set theory, connections with algebra and analysis, forcing axioms and inner model theory. The presence of an unusually large number (19) of young researchers made the meeting especially dynamic
Combinatorial images of sets of reals and semifilter trichotomy
Using a dictionary translating a variety of classical and modern covering
properties into combinatorial properties of continuous images, we get a simple
way to understand the interrelations between these properties in ZFC and in the
realm of the trichotomy axiom for upward closed families of sets of natural
numbers. While it is now known that the answer to the Hurewicz 1927 problem is
positive, it is shown here that semifilter trichotomy implies a negative answer
to a slightly weaker form of this problem.Comment: Small update
Covering the Baire space by families which are not finitely dominating
It is consistent (relative to ZFC) that the union of max{b,g} many families
in the Baire space which are not finitely dominating is not dominating. In
particular, it is consistent that for each nonprincipal ultrafilter U, the
cofinality of the reduced ultrapower w^w/U is greater than max{b,g}. The model
is constructed by oracle chain condition forcing, to which we give a
self-contained introduction.Comment: Small update