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The elementary diagram of a trivial, weakly minimal structure is near model complete
elementary diagram model complete
2015/9/28
We prove that if M is any model of a trivial, weakly minimal theory,
then the elementary diagram T(M) eliminates quantifiers down
to Boolean combinations of certain existential formulas.
On the computability-theoretic complexity of trivial, strongly minimal models
computability-theoretic strongly minimal models
2015/9/28
We show the existence of a trivial, strongly minimal (and thus
uncountably categorical) theory for which the prime model is computable and
each of the other countable models computes 0
00. This res...
Model completeness for trivial, uncountably categorical theories of Morley rank one
uncountably categorical theories Model completeness
2015/9/28
The present paper is a direct continuation of [2], where it is shown that any strongly
minimal trivial theory is model complete after naming constants for a model. In this
paper we show that this re...
Trivial, strongly minimal theories are model complete after naming constants
model complete naming constants
2015/9/25
We prove that if M is any model of a trivial, strongly
minimal theory, then the elementary diagram Th(MM ) is a model
complete LM -theory. We conclude that all countable models of
a trivial, strong...
Borel complexity of complete, first order theories(status report)
first order theories status report
2015/9/25
Borel complexity of complete, first order theories(status report).
The rise and fall of uncountable models.
A Vaught’s conjecture toolbox.
We give a model theoretic proof that if there is a counterexample to Vaught’s conjecture there is a
counterexample such that every model of cardinality ℵ1 is maximal (strengthening a result of ...
Borel completeness of some aleph_0 stable theories
Borel completeness some aleph_0 stable theories
2015/9/25
We study ℵ0-stable theories, and prove that if T either has eniDOP
or is eni-deep, then its class of countable models is Borel complete.
We introduce the notion of λ-Borel completeness and pro...
P-NDOP and P-decompositions of aleph_epsilon saturated models of superstable theories
P-NDOP P-decompositions
2015/9/25
Given a complete, superstable theory, we distinguish a class P of
regular types, typically closed under automorphisms of C and non-
orthogonality. We define the notion of P-NDOP, which is a weakenin...
ω-stable theories: Do uncountable languages matter?
ω-stable theories uncountable languages matter
2015/9/25
ω-stable theories: Do uncountable languages matter?
An old friend revisited: Countable models of ω-stable theories
Countable models ω-stable theories
2015/9/25
We work in the context of ω-stable theories. We obtain a natural,
algebraic equivalent of ENI-NDOP and discuss recent joint proofs with
S. Shelah that if an ω-stable theory has either ENI-DOP or is ...
We characterize the stable theories T for which the saturated
models of T admit decompositions. In particular, we show that
countable, shallow, stable theories with NDOP have this property.
Descriptive set theory and uncountable model theory
Descriptive set theory uncountable model theory
2015/9/25
In the early days of the development of model theory it was considered
natural and was certainly beneficial to assume that the theories under investigation
were in a countable language. The primary ...
Every countable, strictly stable theory either has the Dimensional
Order Property (DOP), is deep, or admits an ‘abelian group witness
to unsuperstability’. To obtain this and other results, we devel...