1,214 research outputs found
Borel sets with large squares
For a cardinal mu we give a sufficient condition (*)_mu (involving ranks
measuring existence of independent sets) for:
[(**)_mu] if a Borel set B subseteq R x R contains a mu-square (i.e. a set of
the form A x A, |A|= mu) then it contains a 2^{aleph_0}-square and even a
perfect square,
and also for
[(***)_mu] if psi in L_{omega_1, omega} has a model of cardinality mu then it
has a model of cardinality continuum generated in a nice, absolute way.
Assuming MA + 2^{aleph_0}> mu for transparency, those three conditions
((*)_mu, (**)_mu and (***)_mu) are equivalent, and by this we get e.g.: for all
alpha= aleph_alpha => not (**)_{aleph_alpha}, and also
min {mu :(*)_mu}, if <2^{aleph_0}, has cofinality aleph_1.
We deal also with Borel rectangles and related model theoretic problems
The complexity of classification problems for models of arithmetic
We observe that the classification problem for countable models of arithmetic
is Borel complete. On the other hand, the classification problems for finitely
generated models of arithmetic and for recursively saturated models of
arithmetic are Borel; we investigate the precise complexity of each of these.
Finally, we show that the classification problem for pairs of recursively
saturated models and for automorphisms of a fixed recursively saturated model
are Borel complete.Comment: 15 page
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
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