Terrence Millar

• Professor Emeritus at University of Wisconsin–Madison

Without Abstract

Recursive model theory is one of the several areas of interaction between recursion theory and model theory. Researchers in both of these classic disciplines benefit from the tools and perspectives of the other. For instance, the cardinality and automorphisms of a structure and the categoricity and existence of prime and saturated models of a theor...

Universal theories with model completions are characterized. A new omitting types theorem is proved. These two results are used to prove the existence of a universal $\aleph_0$-categorical partial order with an interesting embedding property. Other aspects of these results also are considered.

Universal theories with model completions are characterized. A new omitting types theorem is proved. These two results are used to prove the existence of a universal $\aleph_0$-categorical partial order with an interesting embedding property. Other aspects of these results also are considered.

A tame theory is a decidable first-order theory with only countablymany countable models, and all complete types recursive. It is shown here thatthe recursive complexity of countable homogeneous models of tame theories isunbounded in the hyperarithmetic hierarchy.

An Ehrenfeucht theory is a complete first order theory with exactly n countable models up to isomorphism, 1 < n < ω . Numerous results have emerged regarding these theories ([1]–[15]). A general question in model theory is whether or not the number of countable models of a complete theory can be different than the number of countable models of a co...

This paper introduces and investigates a notion that approximates decidability with respect to countable structures. The paper demonstrates that there exists a decidable first order theory with a prime model that is not almost decidable . On the other hand it is proved that if a decidable complete first order theory has only countably many complete...

This paper is concerned with recursive structures and the persistance of an effective notion of categoricity. The terminology and notational conventions are standard. We will devote the rest of this paragraph to a cursory review of some of the assumed conventions. If θ is a formula, then θ k denotes θ if k is zero, and ¬ θ if k is one. If A is a se...

On considere des theories decidables completes, avec seulement des types complets recursifs, qui ont des modeles indecidables denombrables. On etudie une telle theorie qui a seulement un nombre denombrable de modeles denombrables a un isomorphisme pres et en fait pour laquelle le modele saturee denombrable est indecidable

The main result shows that a certain class of theories, the persistently finite, persistently arithmetic theories, have only arithmetic countable models.

Nerode asked if there could be a complete decidable theory with only finitely many countable models up to isomorphism, such that not all of the countable models were decidable. Morley, Lachlan, and Peretyatkin produced examples of such theories. However, all the countable models of those theories were decidable in 0'. The question then arose whethe...

Meeting of the Association for Symbolic Logic, Milwaukee, 1981 - Volume 48 Issue 2 - Jon Barwise, Robert Soare, Terrence Millar

The main result shows that a certain class of theories, the persistently finite, persistently arithmetic theories, have only arithmetic countable models.

Notations, conventions, and definitions. { μ i ∣ i < ω } will be an effective enumeration of all partial recursive μ i { ω → 2. A type of a theory T will be a set of formulas in the language of T , in finitely many free variables, which is consistent with T . A complete type is a maximal type in some fixed number of free variables. A type is recurs...

We prove that for every countable homogeneous model $\mathscr{A}$ such that the set of recursive types of Th(\mathscr{A}$) is \sum^0_2, \mathscr{A}$ is decidable $\operatorname{iff}$ the set of types realized in $\mathscr{A}$ is a $\sum^0_2$ set of recursive types. As a corollary to a lemma, we show that if a complete theory $T$ has a recursively s...

We prove that for every countable homogeneous model Ҩ such that the set of recursive types of Th(Ҩ) is Σ02Ҩ is decidable iff the set of types realized in Ҩ is a Σ02set of recursive types. As a corollary to a lemma, we show that if a complete theory T has a recursively saturated model that is decidable in the degree of T then T has a prime model.

In this paper we investigate the relationship between the number of countable and decidable models of a complete theory. The number of decidable models will be determined in two ways, in §1 with respect to abstract isomorphism type, and in §2 with respect to recursive isomorphism type. Definition 1. A complete theory is ( α , β ) if the number of c...

Fix a countable first order structure A realizing only recursive types. It is known that if A is prime or saturated then it is decidable iff the set of types it realizes is recursively enumerable. A natural conjecture was that the techniques of proof for those two cases could be combined to produce the result for those A that are homogeneous. This...

A well-known result of Vaught's is that no complete theory has exactly two nonisomorphic countable models. The main result of this paper is that there is a complete decidable theory with exactly two nonisomorphic decidable models. A model is decidable if it has a decidable satisfaction predicate. To be more precise, let T be a decidable theory, let...

Photocopy of transcript. Thesis (Ph.D.)--Cornell University, January, 1976. Bibliography: leaves 90-91.