3,654 research outputs found
Even more simple cardinal invariants
Using GCH, we force the following: There are continuum many simple cardinal
characteristics with pairwise different values.Comment: a few changes (minor corrections) from first versio
Club guessing and the universal models
We survey the use of club guessing and other pcf constructs in the context of
showing that a given partially ordered class of objects does not have a
largest, or a universal element
Logical Dreams
We discuss the past and future of set theory, axiom systems and independence
results. We deal in particular with cardinal arithmetic
On Ordinal Invariants in Well Quasi Orders and Finite Antichain Orders
We investigate the ordinal invariants height, length, and width of well quasi
orders (WQO), with particular emphasis on width, an invariant of interest for
the larger class of orders with finite antichain condition (FAC). We show that
the width in the class of FAC orders is completely determined by the width in
the class of WQOs, in the sense that if we know how to calculate the width of
any WQO then we have a procedure to calculate the width of any given FAC order.
We show how the width of WQO orders obtained via some classical constructions
can sometimes be computed in a compositional way. In particular, this allows
proving that every ordinal can be obtained as the width of some WQO poset. One
of the difficult questions is to give a complete formula for the width of
Cartesian products of WQOs. Even the width of the product of two ordinals is
only known through a complex recursive formula. Although we have not given a
complete answer to this question we have advanced the state of knowledge by
considering some more complex special cases and in particular by calculating
the width of certain products containing three factors. In the course of
writing the paper we have discovered that some of the relevant literature was
written on cross-purposes and some of the notions re-discovered several times.
Therefore we also use the occasion to give a unified presentation of the known
results
On cardinal invariants and generators for von Neumann algebras
We demonstrate how virtually all common cardinal invariants associated to a
von Neumann algebra M can be computed from the decomposability number, dec(M),
and the minimal cardinality of a generating set, gen(M). Applications include
the equivalence of the well-known generator problem, "Is every separably-acting
von Neumann algebra singly-generated?", with the formally stronger questions,
"Is every countably-generated von Neumann algebra singly-generated?" and "Is
the gen invariant monotone?" Modulo the generator problem, we determine the
range of the invariant (gen(M), dec(M)), which is mostly governed by the
inequality dec(M) leq c^{gen(M)}.Comment: 22 pages; the main additions are Theorem 3.8 and Section
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