Complete theories with finitely many countable models. II

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. Supported by RFBR grant Nos. 02-01-00258 and 05-01-00411. __________ Translated from Algebra i Logika, Vol. 45, No. 3, pp. 314–353, May–June, 2006.     

A note on countable models of 1-based theories

 We prove that the existence of a nonisolated type having a finite domain and which is orthogonal to ?in a 1-based theory implies that it has a continuum nonisomorphic countable models. Received: 6 March 2000 Published online: 12 July 2002     

Hypergraphs of prime models and distributions of countable models of small theories

Hypergraphs of prime models over realizations of types in small theories are defined. On the basis of graph structures of models of small theories, hierarchies of sets in these hypergraphs, revealing structural connections in countable models of small theories, are established. The key role of graph-theoretic objects in constructions of Ehrenfeucht theories is proved. Using hypergraph constructions, a classification of complete first-order theories with finite Rudin–Keisler preorders is generalized to the class of all small theories.     

Stable theories, pseudoplanes and the number of countable models

We prove that if T is a stable theory with only a finite number (>1) of countable models, then T contains a type-definable pseudoplane. We also show that for any stable theory T either T contains a type-definable pseudoplane or T is weakly normal (in the sense of [9]).     

Iterated expansions of models for countable theories and their applications

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 -expansion are proved, and we settle the question of whether or not -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.Translated fromAlgebra i Logika, Vol. 34, No. 6, pp. 623-645, November-December, 1995.Supported by RFFR grant No. 93-01-01506.     

On the number of countable models of complete theories with finite Rudin-Keisler preorders

The aim of this article is to generalize the classification of complete theories with finitely many countable models with respect to two principal characteristics, Rudin-Keisler preorders and the distribution functions of the number of limit models, to an arbitrary case with a finite Rudin-Keisler preorder. We establish that the same characteristics play a crucial role in the case we consider. We prove the compatibility of arbitrary finite Rudin-Keisler preorders with arbitrary distribution functions f satisfying the condition rang f?ω∪{ω, 2^ω}.     

Theories with constants and three countable models

We prove that a countable, complete, first-order theory with infinite dcl() and precisely three non-isomorphic countable models interprets a variant of Ehrenfeucht’s or Peretyatkin’s example.     

Superstable groups

We initiate a study of superstable groups, generalizing previous work of Zil'ber and Cherlin. After having introduced the various tools of stability and model theory needed for that purpose we prove a general ‘Indecomposability Theorem’ and apply them to prove:(1) the definability of many subgroups of superstable groups (which has the consequence that, for superstable groups, “to be simple” is a first-order property);(2) the existence of ‘large’ abelian subgroups of all superstable groups; this allows us for example to give a transparent proof to the theorem of Cherlin stating that superstable division rings are commutative.This study of superstable groups is continued in [4].     

On Banach spaces with few spreading models

If the set of spreading models of a Banach space is countable (up to equivalence), then it cannot contain a strictly increasing infinite chain of spreading models generated by normalized weakly null sequences. Moreover, such a space must have a spreading model which is `close' to or for some .

     

Simple groups and the number of countable models

Let T be a complete, superstable theory with fewer than ${2^{\aleph_{0}}}$ countable models. Assuming that generic types of infinite, simple groups definable in T ^eq are sufficiently non-isolated we prove that ω ^ω is the strict upper bound for the Lascar rank of T.     

The classification of countable models of set theory

We study the complexity of the classification problem for countable models of set theory ($\mathsf{ZFC}$). We prove that the classification of arbitrary countable models of $\mathsf{ZFC}$ is Borel complete, meaning that it is as complex as it can conceivably be. We then give partial results concerning the classification of countable well-founded models of $\mathsf{ZFC}$.     

Superstable semigroups of operators

In [NR] the authors introduced the notion of superstable operators on a Banach space E using ultrapowers E_u of E. In [HR] this notion was extended to strongly continuous one-parameter semigroups again by means of ultrapowers.It is the aim of the present paper to give an equivalent intrinsic definition of superstability (without the reference to ultrapowers). This definition allows us to improve the results of [NR] as well as of [HR]. We apply our results to semigroups of positive linear operators on Banach lattices and C^*-algebras, respectively.     

Superstable fields and groups

We prove an indecomposability theorem for connected stable groups. Using this theorem we prove that all infinite superstable fields are algebraically closed, and we extend known results for ω-stable groups of Morley rank at most 3 to the corresponding class of superstable groups (Note: The logical notion of stability is unrelated to the notion of stability in finit group theory).