345 research outputs found
On constructions with -cardinals
We propose developing the theory of consequences of morasses relevant in
mathematical applications in the language alternative to the usual one,
replacing commonly used structures by families of sets originating with
Velleman's neat simplified morasses called -cardinals. The theory of related
trees, gaps, colorings of pairs and forcing notions is reformulated and
sketched from a unifying point of view with the focus on the applicability to
constructions of mathematical structures like Boolean algebras, Banach spaces
or compact spaces.
A new result which we obtain as a side product is the consistency of the
existence of a function
with the
appropriate -version of property for regular
satisfying .Comment: Minor correction
Existence of optimal ultrafilters and the fundamental complexity of simple theories
In the first edition of Classification Theory, the second author
characterized the stable theories in terms of saturation of ultrapowers. Prior
to this theorem, stability had already been defined in terms of counting types,
and the unstable formula theorem was known. A contribution of the ultrapower
characterization was that it involved sorting out the global theory, and
introducing nonforking, seminal for the development of stability theory. Prior
to the present paper, there had been no such characterization of an unstable
class. In the present paper, we first establish the existence of so-called
optimal ultrafilters on Boolean algebras, which are to simple theories as
Keisler's good ultrafilters are to all theories. Then, assuming a supercompact
cardinal, we characterize the simple theories in terms of saturation of
ultrapowers. To do so, we lay the groundwork for analyzing the global structure
of simple theories, in ZFC, via complexity of certain amalgamation patterns.
This brings into focus a fundamental complexity in simple unstable theories
having no real analogue in stability.Comment: The revisions aim to separate the set theoretic and model theoretic
aspects of the paper to make it accessible to readers interested primarily in
one side. We thank the anonymous referee for many thoughtful comment
On what I do not understand (and have something to say): Part I
This is a non-standard paper, containing some problems in set theory I have
in various degrees been interested in. Sometimes with a discussion on what I
have to say; sometimes, of what makes them interesting to me, sometimes the
problems are presented with a discussion of how I have tried to solve them, and
sometimes with failed tries, anecdote and opinion. So the discussion is quite
personal, in other words, egocentric and somewhat accidental. As we discuss
many problems, history and side references are erratic, usually kept at a
minimum (``see ... '' means: see the references there and possibly the paper
itself).
The base were lectures in Rutgers Fall'97 and reflect my knowledge then. The
other half, concentrating on model theory, will subsequently appear
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
An Algebraic and Logical approach to continuous images
Continuous mappings between compact Hausdorff spaces can be studied using
homomorphisms between algebraic structures (lattices, Boolean algebras)
associated with the spaces. This gives us more tools with which to tackle
problems about these continuous mappings -- also tools from Model Theory. We
illustrate by showing that the \v{C}ech-Stone remainder has a
universality property akin to that of ; a theorem of Ma\'ckowiak and
Tymchatyn implies it own generalization to non-metric continua; and certain
concrete compact spaces need not be continuous images of .Comment: Notes from a series of lectures at
http://www.cts.cuni.cz/events/ws/2002/ws2002.htm, the 30th Winter School on
Abstract Analysis 2002-05-02: corrected version after referee's repor
Dense ideals and cardinal arithmetic
From large cardinals we show the consistency of normal, fine,
-complete -dense ideals on for
successor . We explore the interplay between dense ideals, cardinal
arithmetic, and squares, answering some open questions of Foreman
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