Limit Models of Ehrenfeucht Theories

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 complete consistent extension of the theory by finitely many constant symbols. Examples are known of Ehrenfeucht theories that have complete extensions by finitely many constant symbols such that the extensions fail to be Ehrenfeucht ([4], [8], [13]). These examples are easily modified to allow finite increases in the number of countable models.This paper contains examples in the other direction—complete theories that have consistent extensions by finitely many constant symbols such that the extensions have fewer countable models. This answers affirmatively a question raised by, among others, Peretyat'kin [8]. The first example will be an Ehrenfeucht theory with exactly four countable models with an extension by a constant symbol that has only three countable models. The second example will be a complete theory that is not Ehrenfeucht, but which has an extension by a constant symbol that is Ehrenfeucht. The notational conventions for this paper are standard.Peretyat'kin introduced the theory of a dense binary branching tree with a meet operator [7]. Dense ω-branching trees have also proven useful [5], [11]. Both of the Theories that will be constructed make use of dense ω-branching trees.

Previously, we obtained a syntactic characterization for the class of complete theories with finitely many pairwise non-isomorphic countable models [1]. The most essential part of that characterization extends to Ehrenfeucht theories (i.e., those having finitely many (but more than 1) pairwise non-isomorphic countable models). As the basic parameters defining a finite number of countable models, Rudin-Keisler quasiorders are treated as well as distribution functions defining the number of limit models for equivalence classes w.r.t. these quasiorders. Here, we argue to state that all possible parameters given in the characterization theorem in [1] are realizable. Also, we describe Rudin-Keisler quasiorders in arbitrary small theories. The construction of models of Ehrenfeucht theories with which we come up in the paper is based on using powerful digraphs which, along with powerful types in Ehrenfeucht theories, always locally exist in saturated models of these theories.

We study complexity of the index set of countably categorical theories and Ehrenfeucht theories in finite languages.

In this paper we investigate computable models of $\aleph_1$-categorical theories and Ehrenfeucht theories. For instance, we give an example of an $\aleph_1$-categorical but not $\aleph_0$-categorical theory $T$ such that all the countable models of $T$ except its prime model have computable presentations. We also show that there exists an $\aleph_1$-categorical but not $\aleph_0$-categorical theory $T$ such that all the countable models of $T$ except the saturated model, have computable presentations.

We study the problem of expanding and extending the structure of a stable powerful digraph to the structure of a stable Ehrenfeucht theory. We define the concepts of type unstability and type strict order property. We establish the presence of the type strict order property for every acyclic graph structure with an infinite chain. The simplest form of expansion of a powerful digraph to the structure of an Ehrenfeucht theory is the expansion with a 1-inessential ordered coloring and locally graph ∃-definable many-placed relations, which enable us to mutually realize nonprincipal types; we prove that this expansion is incapable of keeping the structure in the class of stable structures, and moreover, by the type strict order property it generates the first-order definable strict order property. We define the concept of a locally countably categorical theory (LCC theory) and prove that given the list p1(x), ..., pn(x) of all nonprincipal 1-types in an LCC theory, if all types r(x1, ..., xm) containing \( p_{i_1 } \) (x1) ∪ ... ∪ \( p_{i_m } \)(xm) are dominated by some type q then q is a powerful type.

It is known that the class of Ehrenfeucht theories admits a syntactical characterization and that a finite (Rudin-Keisler) pre-ordering and a function mapping this pre-ordering to naturals play the role of parameters in this characterization. In the article, we construct for any finite linear ordering L, a hereditary decidable Ehrenfeucht theory T possessing L as its Rudin-Keisler pre-ordering. Also, we discuss decidable and computable models of such theories.

In this paper we concentrate on open problems in two directions in the development of the theory of constructive algebraic systems. The first direction deals with universal algebras whose positive open diagrams can be computably enumerated. These algebras are called positive algebras. Here we emphasize the interplay between universal algebra and computability theory. We propose a systematic study of positive algebras as a new direction in the development of the theory of constructive algebraic systems. The second direction concerns the traditional topics in constructive model theory. First we propose the study of constructive models of theories with few models such as countably categorical theories, uncountably categorical theories, and Ehrenfeucht theories. Next, we propose the study of computable isomorphisms and computable dimensions of such models. We also discuss issues related to the computability-theoretic complexity of relations in constructive algebraic sys-

Constant or nonessential extensions of elementary theories provide a productive tool for the study and structural description of models of these theories, which is widely used in Model Theory and its applications, both for various stable and ordered theories, countable and uncountable theories, algebraic, geometric and relational structures and theories. Families of constants are used in Henkin’s classical construction of model building for consistent families of formulas, for the classification of uncountable and countable models of complete theories, and for some dynamic possibilities of countable spectra of ordered Ehrenfeucht theories. The paper describes the possibilities of ranks and degrees for families of constant extensions of theories. Rank links are established for families of theories with CantorBendixson ranks for given theories. It is shown that the 𝑒-minimality of a family of constant expansions of the theory is equivalent to the existence and uniqueness of a nonprincipal type with a given number of variables. In particular, for strongly minimal theories this means that the non-principal 1-type is unique over an appropriate tuple. Relations between 𝑒-spectra of families of constant expansions of theories and ranks and degrees are established. A model-theoretic characterization of the existence of the least generating set is obtained. It is also proved that any inessential finite expansion of an o-minimal Ehrenfeucht theory preserves the Ehrenfeucht property, and this is true for constant expansions of dense spherically ordered theories. For the expansions under consideration, the dynamics of the values of countable spectra is described.

An expansion of a countable model by relations for incomplete types realized in the model is constructed. Theories of models obtained by iterating the expansion defined over countable ordinals are investigated. Theorems concerning the atomicity of countable models in a suitable α-expansion are proved, and we settle the question of whether or not α-expansions have atomic models. A theorem on the realization and omission of generalized types is presented. The resulis obtained are then used to give a direct proof of a theorem of Morley on the number of countable models and to state that Ehrenfeucht theories have finite type rank.

We construct an example of a theory with a finite (greater than one) number of isomorphism types of countable models such that its prime and saturated models have computable presentations and there exists a model which lacks in such.

A decidable Ehrenfeucht theory with exactly two hyperarithmetic models

We describe Rudin–Keisler preorders and distribution functions of numbers of limit models for disjoint unions of Ehrenfeucht theories. We also find decomposition formulas for these distributions.

The index set of a computable structure $$\mathcal{A}$$ is the set of indices for computable copies of $$\mathcal{A}$$ . We determine complexity of the index sets of various mathematically interesting structures including different finite structures, ℚ-vector spaces, Archimedean real-closed ordered fields, reduced Abelian p-groups of length less than ω2, and models of the original Ehrenfeucht theory. The index sets for these structures all turn out to be m-complete Π n 0 , d-Σ n 0 , or Σ n 0 , for various n. In each case the calculation involves finding an optimal sentence (i.e., one of simplest form) that describes the structure. The form of the sentence (computable Πn, d-Σn, or Σn) yields a bound on the complexity of the index set. Whenever we show m-completeness of the index set, we know that the sentence is optimal. For some structures, the first sentence that comes to mind is not optimal, and another sentence of simpler form is shown to serve the purpose. For some of the groups, this involves Ramsey’s theory.

In the paper, we investigate Ehrenfeucht theories, that is, theories which have nitely many countable models but which are not countably categorical.More precisely, we count all possible numbers of countable models of the theory DMT of dense meettrees expanded by several sequences of constants including decreasing ones and by unary predicates with nite realizations.Also, we study the realizations of models over a certain set of formulas based on the Rudin-Keisler preorders on models.

We study all possible constant expansions of the structure of the dense meet-tree ⟨М; <, П⟩ [3]. Here, a dense meet-tree is a lower semilattice without the least and greatest elements. An example of this structure with the constant expansion is a theory that has exactly three pairwise non-isomorphic countable models [6], which is a good example in the context of Ehrenfeucht theories. We study all possible constant expansions of the structure of the dense meet-tree by using a general theory of classification of countable models of complete theories [7], as well as the description of the specificity for the theory of a dense-meet tree, namely, some distributions of countable models of these theories in terms of Rudin– Keisler preorders and distribution functions of numbers of limit models. In this paper, we give a new proof of the theorem that the dense meet-tree theory is countable categorical and complete, which was originally proved by Peretyat’kin. Also, this theory admits quantifier elimination since complete types are forced by a set of quantifier-free formulas, and this leads to the fact that it is decidable