In Bue93b] we proved Vaught's conjecture for all superstable theories of nite rank; that is, such a theory has countably many or continuum many countable models. While this proof does settle Vaught's conjecture for unidimensional theories a sharper result can be obtained for these theories and in places the proof can be simpliied. Let T be a properly unidimensional theory with < 2 @ 0 many coun...

In 2004 Csima, Hirschfeldt, Knight, and Soare [1] showed that a set A ≤T 0′ is nonlow2 if and only if A is prime bounding, i.e. for every complete atomic decidable theory T , there is a prime model M computable in A. The authors presented nine seemingly unrelated predicates of a set A, and showed that they are equivalent for ∆2 sets. Some of these predicates, such as prime bounding, and others ...

We prove that many properties and invariants of definable groups in NIP theories, such as definable amenability, G/G00, etc., are preserved when passing to the theory of the Shelah expansion by externally definable sets, Mext, of a model M . In the light of these results we continue the study of the “definable topological dynamics” of groups in NIP theories. In particular we prove the Ellis gro...

Here, T is the time of an event and Z a p-dimensional vector of covariates, φ a square integrable function, and F(t | z) = P (T ≤ t | Z = z) and H( dz) = P (Z ∈ dz) denote the conditional survival distribution and the marginal law of Z , respectively. For example, θ(I {t > t∗}) is the marginal survival probability at time t∗, θ(I {t > t∗, z1 > z∗ 1}) the bivariate distribution at (t∗, z∗ 1) (Ak...

Theorem 1.1 (Hrushovski’s Theorem 3.5). Let G be a definable group, X a definable subset, and G̃ be the ∨-definable group generated by X. Suppose all this data is defined over a model M . Suppose μ is an M-invariant translation-invariant ideal on subsets of G̃. Let q be a wide type over M , contained in G̃. Identify q with its ∧-definable set of realizations. Suppose (F) there exist two realizatio...

In this paper we prove that if κ is a cardinal in L[0]], then there is an inner model M such that M |= (Vκ,∈) has no elementary end extension. In particular if 0] exists, then weak compactness is never downwards absolute. We complement the result with a lemma stating that any cardinal greater than א1 of uncountable cofinality in L[0]] is Mahlo in every strict inner model of L[0]].

The following assertion is an exercise in Lecture 4. Let ψ be a complete sentence in Lω1,ω in a countable language L. Then there is a countable language L ′ extending L and a first order L′-theory T such that reduct is a 1-1 map from the atomic models of T onto the models of ψ. So in particular, any complete sentence of Lω1,ω can be replaced (for spectrum purposes) by considering the atomic mod...

This paper presents a Bayesian interpretation of maximum entropy image reconstruction and shows that exp(αS(/, m)), where S(f,m) is the entropy of image / relative to model m, is the only consistent prior probability distribution for positive, additive images. It also leads to a natural choice for the regularizing parameter α, that supersedes the traditional practice of setting χ = N. The new c...

In this paper we answer a question of Friedman, providing an ω-separable model M of the λβη-calculus. There therefore exists an α-separable model for any α ≥ 0. The model M permits no non-trivial enrichment as a partial order; neither does it permit an enrichment as a category with an initial object. The open term model embeds in M: by way of contrast we provide a model which cannot embed in an...

We study the minimal enumeration degree (e-degree) problem in models of fragments of Peano arithmetic (PA) and prove the following results: In any model M of Σ2 induction, there is a minimal enumeration degree if and only if M is a nonstandard model. Furthermore, any cut in such a model has minimal e-degree. By contrast, this phenomenon fails in the absence of Σ2 induction. In fact, whether eve...