76 research outputs found
Laver and set theory
In this commemorative article, the work of Richard Laver is surveyed in its full range and extent.Accepted manuscrip
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
Kurepa trees and the failure of the Galvin property
We force the existence of a non-trivial -complete ultrafilter over
which fails to satisfy the Galvin property. This answers a question
asked by the first author and Moti Gitik
Galvin's property at large cardinals and the axiom of determinancy
In the first part of this paper, we explore the possibility for a very large
cardinal to carry a -complete ultrafilter without Galvin's
property. In this context, we prove the consistency of every ground model
-complete ultrafilter extends to a non-Galvin one. Oppositely, it is
also consistent that every ground model -complete ultrafilter extends
to a -point ultrafilter, hence to another one satisfying Galvin's property.
We also study Galvin's property at large cardinals in the choiceless context,
especially under \textsf{AD}. Finally, we apply this property to a classical
pro\-blem in partition calculus by proving the relation
under
``\textsf{AD}+'' for unboundedly many below
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