105 research outputs found
On the complexity of Hamel bases of infinite dimensional Banach spaces
We call a subset S of a topological vector space V linearly Borel, if for
every finite number n, the set of all linear combinations of S of length n is a
Borel subset of V. It will be shown that a Hamel base of an infinite
dimensional Banach space can never be linearly Borel. This answers a question
of Anatolij Plichko
Ramseyan ultrafilters
We investigate families of partitions of omega which are related to special
coideals, so-called happy families, and give a dual form of Ramsey ultrafilters
in terms of partitions. The combinatorial properties of these
partition-ultrafilters, which we call Ramseyan ultrafilters, are similar to
those of Ramsey ultrafilters. For example it will be shown that dual Mathias
forcing restricted to a Ramseyan ultrafilter has the same features as Mathias
forcing restricted to a Ramsey ultrafilter. Further we introduce an ordering on
the set of partition-filters and consider the dual form of some cardinal
characteristics of the continuum
Consequences of arithmetic for set theory
In this paper, we consider certain cardinals in ZF (set theory without AC,
the Axiom of Choice). In ZFC (set theory with AC), given any cardinals C and D,
either C <= D or D <= C. However, in ZF this is no longer so. For a given
infinite set A consider Seq(A), the set of all sequences of A without
repetition. We compare |Seq(A)|, the cardinality of this set, to |P(A)|, the
cardinality of the power set of A.
What is provable about these two cardinals in ZF? The main result of this
paper is that
ZF |- for all A: |Seq(A)| not= |P(A)|
and we show that this is the best possible result.
Furthermore, it is provable in ZF that if B is an infinite set, then
|fin(B)|<|P(B)|, even though the existence for some infinite set B^* of a
function f from fin(B^*) onto P(B^*) is consistent with ZF
Techniques for approaching the dual Ramsey property in the projective hierarchy
We define the dualizations of objects and concepts which are essential for
investigating the Ramsey property in the first levels of the projective
hierarchy, prove a forcing equivalence theorem for dual Mathias forcing and
dual Laver forcing, and show that the Harrington-Kechris techniques for proving
the Ramsey property from determinacy work in the dualized case as well
Ultrafilter spaces on the semilattice of partitions
The Stone-Cech compactification of the natural numbers bN, or equivalently,
the space of ultrafilters on the subsets of omega, is a well-studied space with
interesting properties. If one replaces the subsets of omega by partitions of
omega, one can define corresponding, non-homeomorphic spaces of partition
ultrafilters. It will be shown that these spaces still have some of the nice
properties of bN, even though none is homeomorphic to bN. Further, in a
particular space, the minimal height of a tree pi-base and P-points are
investigated
Relations between some cardinals in the absence of the Axiom of Choice
If we assume the axiom of choice, then every two cardinal numbers are
comparable. In the absence of the axiom of choice, this is no longer so. For a
few cardinalities related to an arbitrary infinite set, we will give all the
possible relationships between them, where possible means that the relationship
is consistent with the axioms of set theory. Further we investigate the
relationships between some other cardinal numbers in specific permutation
models and give some results provable without using the axiom of choice
Mathias absoluteness and the Ramsey property
In this article we give a forcing characterization for the Ramsey property of -Sets of reals. This research was motivated by the well-known forcing characterizations for Lebesgue measurability and the Baire property of -sets of reals. Further we will show the relationship between higher degrees of forcing absoluteness and the Ramsey property of projective sets of real
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