Semi-minimal theories and categoricity

1975 ◽  
Vol 40 (3) ◽  
pp. 419-438 ◽  
Author(s):  
Daniel Andler
Keyword(s):  
Simple Proof ◽  
Minimal Theory ◽  
Two Stages ◽  
Countable Theory ◽  
Countable Models

The study of countable theories categorical in some uncountable power was initiated by Łoś and Vaught and developed in two stages. First, Morley proved (1962) that a countable theory categorical in some uncountable power is categorical in every uncountable power, a conjecture of Łoś. Second, Baldwin and Lachlan confirmed (1969) Vaught's conjecture that a countable theory categorical in some uncountable power has either one or countably many isomorphism types of countable models. That result was obtained by pursuing a line of research developed by Marsh (1966). For certain well-behaved theories, which he called strongly minimal, Marsh's method yielded a simple proof of Łoś's conjecture and settled Vaught's conjecture.In recent years efforts have been made to extend these results to uncountable theories. The generalized Łoś conjecture states that a theory T categorical in some power greater than ∣T∣ is categorical in every such power. It was settled by Shelah (1970). Shelah then raised the question of the models in power ∣T∣ = ℵα of a theory T categorical in ∣T∣+, conjecturing in [S3] that there are exactly ∣α∣ + ℵ0 such models, up to isomorphism. This conjecture provided the initial motivation for the present work. We define and study semi-minimal theories analogous in some ways to Marsh's strongly minimal (countable) theories. We describe the models of a semi-minimal theory T which contain an infinite indiscernible set. Besides throwing some light on Shelah's conjecture, our method gives simple proofs of the Łoś conjecture and of the Morley conjecture on categoricity in ∣T∣, in the case of a semi-minimal theory T. Other results as well as some examples are provided.

Why some people are excited by Vaught's conjecture

1985 ◽  
Vol 50 (4) ◽  
pp. 973-982 ◽  
Author(s):  
Daniel Lascar
Keyword(s):  
Model Theory ◽  
Continuum Hypothesis ◽  
Complete Theory ◽  
Scott Sentences ◽  
Countable Theory ◽  
The Continuum ◽  
Countable Models

§I. In 1961, R. L. Vaught ([V]) asked if one could prove, without the continuum hypothesis, that there exists a countable complete theory with exactly ℵ1 isomorphism types of countable models. The following statement is known as Vaught conjecture:Let T be a countable theory. If T has uncountably many countable models, then T hascountable models.More than twenty years later, this question is still open. Many papers have been written on the question: see for example [HM], [M1], [M2] and [St]. In the opinion of many people, it is a major problem in model theory.Of course, I cannot say what Vaught had in mind when he asked the question. I just want to explain here what meaning I personally see to this problem. In particular, I will not speak about the topological Vaught conjecture, which is quite another issue.I suppose that the first question I shall have to face is the following: “Why on earth are you interested in the number of countable models—particularly since the whole question disappears if we assume the continuum hypothesis?” The answer is simply that I am not interested in the number of countable models, nor in the number of models in any cardinality, as a matter of fact. An explanation is due here; it will be a little technical and it will rest upon two names: Scott (sentences) and Morley (theorem).

On the number of countable homogeneous models

1983 ◽  
Vol 48 (3) ◽  
pp. 539-541 ◽  
Author(s):  
Libo Lo
Keyword(s):  
Homogeneous Model ◽  
Countable Model ◽  
The Other ◽  
Reduced Model ◽  
Elementary Extension ◽  
Other Hand ◽  
Homogeneous Models ◽  
Countable Theory ◽  
Countable Models

The number of homogeneous models has been studied in [1] and other papers. But the number of countable homogeneous models of a countable theory T is not determined when dropping the GCH. Morley in [2] proves that if a countable theory T has more than ℵ1 nonisomorphic countable models, then it has such models. He conjectures that if a countable theory T has more than ℵ0 nonisomorphic countable models, then it has such models. In this paper we show that if a countable theory T has more than ℵ0 nonisomorphic countable homogeneous models, then it has such models.We adopt the conventions in [1]–[3]. Throughout the paper T is a theory and the language of T is denoted by L which is countable.Lemma 1. If a theory T has more than ℵ0types, then T hasnonisomorphic countable homogeneous models.Proof. Suppose that T has more than ℵ0 types. From [2, Corollary 2.4] T has types. Let σ be a Ttype with n variables, and T′ = T ⋃ {σ(c1, …, cn)}, where c1, …, cn are new constants. T′ is consistent and has a countable model (, a1, …, an). From [3, Theorem 3.2.8] the reduced model has a countable homogeneous elementary extension . σ is realized in . This shows that every type σ is realized in at least one countable homogeneous model of T. But each countable model can realize at most ℵ0 types. Hence T has at least countable homogeneous models. On the other hand, a countable theory can have at most nonisomorphic countable models. Hence the number of nonisomorphic countable homogeneous models of T is .In the following, we shall use the languages Lα (α = 0, 1, 2) defined in [2]. We give a brief description of them. For a countable theory T, let K be the class of all models of T. L = L0 is countable.

End extensions and numbers of countable models

1978 ◽  
Vol 43 (3) ◽  
pp. 550-562 ◽  
Author(s):  
Saharon Shelah
Keyword(s):  
Infinite Order ◽  
Universal Model ◽  
Skolem Functions ◽  
Countable Theory ◽  
Countable Models

AbstractWe prove that every model of T = Th(ω, <,…) (T countable) has an end extension; and that every countable theory with an infinite order and Skolem functions has nonisomorphic countable models; and that if every model of T has an end extension, then every ∣T∣-universal model of T has an end extension definable with parameters.

Theories with exactly three countable models and theories with algebraic prime models

1980 ◽  
Vol 45 (2) ◽  
pp. 302-310 ◽  
Author(s):  
Anand Pillay
Keyword(s):  
Complete Theory ◽  
Uniform Bound ◽  
Bounded Degree ◽  
Definable Set ◽  
Countable Theory ◽  
Uniformly Bounded ◽  
Pathological Case ◽  
Countable Models

We prove first that if T is a countable complete theory with n(T), the number of countable models of T, equal to three, then T is similar to the Ehrenfeucht example of such a theory. Woodrow [4] showed that if T is in the same language as the Ehrenfeucht example, T has elimination of quantifiers, and n(T) = 3 then T is very much like this example. All known examples of theories T with n(T) finite and greater than one are based on the Ehrenfeucht example. We feel that such theories are a pathological case. Our second theorem strengthens the main result of [2]. The theorem in the present paper says that if T is a countable theory which has a model in which all the elements of some infinite definable set are algebraic of uniformly bounded degree, then n(T) ≥ 4. It is known [3] that if n(T) > 1, then n(T) > 3, so our result is the first nontrivial step towards proving that n(T) ≥ ℵ0. We would also like, of course, to prove the result without the uniform bound on the finite degrees of the elements in the subset.Theorem 2.1 is included in the author's Ph. D. thesis, as is a weaker version of Theorem 3.7. Thanks are due to Harry Simmons for his suggestions concerning the presentation of the material, and to Wilfrid Hodges for his advice while I was a Ph. D. student.

On recursively enumerable and arithmetic models of set theory

1958 ◽  
Vol 23 (4) ◽  
pp. 408-416 ◽  
Author(s):  
Michael O. Rabin
Keyword(s):  
Binary Relation ◽  
Set Theory ◽  
Simple Proof ◽  
Side Product ◽  
Logical Constant ◽  
Logical Constants ◽  
Recursive Models ◽  
Recursively Enumerable ◽  
Models Of Set Theory ◽  
Countable Models

In this note we shall prove a certain relative recursiveness lemma concerning countable models of set theory (Lemma 5). From this lemma will follow two results about special types of such models.Kreisel [5] and Mostowski [6] have shown that certain (finitely axiomatized) systems of set theory, formulated by means of the ϵ relation and certain additional non-logical constants, do not possess recursive models. Their purpose in doing this was to construct consistent sentences without recursive models. As a first corollary of Lemma 5, we obtain a very simple proof, not involving any formal constructions within the system of the notions of truth and satisfiability, of an extension of the Kreisel-Mostowski theorems. Namely, set theory with the single non-logical constant ϵ does not possess any recursively enumerable model. Thus we get, as a side product, an easy example of a consistent sentence containing a single binary relation which does not possess any recursively enumerable model; this sentence being the conjunction of the (finitely many) axioms of set theory.

Number of countable models

1978 ◽  
Vol 43 (3) ◽  
pp. 492-496 ◽  
Author(s):  
Anand Pillay
Keyword(s):  
Finite Number ◽  
Complete Theory ◽  
Prime Model ◽  
Categorical Theory ◽  
Definable Subset ◽  
Algebraic Elements ◽  
Minimal Prime ◽  
Countable Theory ◽  
The Universe ◽  
Countable Models

We prove that a countable complete theory whose prime model has an infinite definable subset, all of whose elements are named, has at least four countable models up to isomorphism. The motivation for this is the conjecture that a countable theory with a minimal model has infinitely many countable models. In this connection we first prove that a minimal prime model A has an expansion by a finite number of constants A′ such that the set of algebraic elements of A′ contains an infinite definable subset.We note that our main conjecture strengthens the Baldwin–Lachlan theorem. We also note that due to Vaught's result that a countable theory cannot have exactly two countable models, the weakest possible nontrivial result for a non-ℵ0-categorical theory is that it has at least four countable models.§1. Notation and preliminaries. Our notation follows Chang and Keisler [1], except that we denote models by A, B, etc. We use the same symbol to refer to the universe of a model. Models we refer to are always in a countable language. For T a countable complete theory we let n(T) be the number of countable models of T up to isomorphism. ∃n means ‘there are exactly n’.

Countable models of multidimensional ℵ0-stable theories

1983 ◽  
Vol 48 (2) ◽  
pp. 377-383 ◽  
Author(s):  
Elisabeth Bouscaren
Keyword(s):  
Satisfy Condition ◽  
Strongly Regular ◽  
Countable Theory ◽  
Additional Assumptions ◽  
Countable Models

The notations and the definitions used here are identical to those in [B.L.]. We also assume that the reader is familiar with the properties of the Rudin-Keisler order on types [L.].Let T denote an ℵ0-stable countable theory. The main result in [B.L.] is: All countable models of T are almost homogeneous if and only if T satisfies(*) For all models M of T and all ā ∈ M, if p ∈ S1(ā) is strongly regular multidimensional, then Dim(p; M) ≥ ℵ0.In this paper we investigate the case of a theory T which does not satisfy condition (*) and, under certain additional assumptions, we construct nonisomorphic and non-almost-homogeneous countable models. The same type of construction has been used before to show that if T is multidimensional, then for all α ≥ 1, T has at least nonisomorphic models of cardinality ℵα [La.], [Sh.]. As a corollary of our main theorem (Theorem 6) and of the previous result in [B.L.], we prove Vaught's Conjecture (and, in fact, Martin's Strong Conjecture) for theories T with αT finite.Although we are here interested in countable models, we can also note that our construction proves that theories satisfying the assumptions in Theorem 6 have at least nonisomorphic models of cardinality ℵα.

scholarly journals THE BOREL COMPLEXITY OF ISOMORPHISM FOR O-MINIMAL THEORIES

2017 ◽  
Vol 82 (2) ◽  
pp. 453-473 ◽  
Author(s):  
RICHARD RAST ◽  
DAVENDER SINGH SAHOTA
Keyword(s):  
Lower Bounds ◽  
General Model ◽  
Isomorphism Problem ◽  
Linear Orders ◽  
Minimal Theory ◽  
Borel Complexity ◽  
Image Position ◽  
Countable Models

AbstractGiven a countable o-minimal theory T, we characterize the Borel complexity of isomorphism for countable models of T up to two model-theoretic invariants. If T admits a nonsimple type, then it is shown to be Borel complete by embedding the isomorphism problem for linear orders into the isomorphism problem for models of T. This is done by constructing models with specific linear orders in the tail of the Archimedean ladder of a suitable nonsimple type.If the theory admits no nonsimple types, then we use Mayer’s characterization of isomorphism for such theories to compute invariants for countable models. If the theory is small, then the invariant is real-valued, and therefore its isomorphism relation is smooth. If not, the invariant corresponds to a countable set of reals, and therefore the isomorphism relation is Borel equivalent to F2.Combining these two results, we conclude that $\left( {{\rm{Mod}}\left( T \right), \cong } \right)$ is either maximally complicated or maximally uncomplicated (subject to completely general model-theoretic lower bounds based on the number of types and the number of countable models).

Vaught's conjecture for o-minimal theories

1988 ◽  
Vol 53 (1) ◽  
pp. 146-159 ◽  
Author(s):  
Laura L. Mayer
Keyword(s):  
Structure Theorem ◽  
Classification Theory ◽  
Real Closed Fields ◽  
Minimal Theory ◽  
Definable Subset ◽  
Closed Fields ◽  
The Universe ◽  
Countable Models

The notion of o-minimality was formulated by Pillay and Steinhorn [PS] building on work of van den Dries [D]. Roughly speaking, a theory T is o-minimal if whenever M is a model of T then M is linearly ordered and every definable subset of the universe of M consists of finitely many intervals and points. The theory of real closed fields is an example of an o-minimal theory.We examine the structure of the countable models for T, T an arbitrary o-minimal theory (in a countable language). We completely characterize these models, provided that T does not have 2ω countable models. This proviso (viz. that T has fewer than 2ω countable models) is in the tradition of classification theory: given a cardinal α, if T has the maximum possible number of models of size α, i.e. 2α, then no structure theorem is expected (cf. [Sh1]).O-minimality is introduced in §1. §1 also contains conventions and definitions, including the definitions of cut and noncut. Cuts and noncuts constitute the nonisolated types over a set.In §2 we study a notion of independence for sets of nonisolated types and the corresponding notion of dimension.In §3 we define what it means for a nonisolated type to be simple. Such types generalize the so-called “components” in Pillay and Steinhorn's analysis of ω-categorical o-minimal theories [PS]. We show that if there is a nonisolated type which is not simple then T has 2ω countable models.

Altered Fine Structure of B Lymphocytes Correlated with Defective Terminal Differentiation

1973 ◽  
Vol 31 ◽  
pp. 386-387
Author(s):  
Dale E. Bockman ◽  
L. Y. Frank Wu ◽  
Alexander R. Lawton ◽  
Max D. Cooper
Keyword(s):  
Immune Deficiency ◽  
B Lymphocytes ◽  
Plasma Cells ◽  
Present Report ◽  
Specific Antigen ◽  
Terminal Differentiation ◽  
Second Stage ◽  
Two Stages ◽  
Deficiency Diseases ◽  
Cell Line Generation

B-lymphocytes normally synthesize small amounts of immunoglobulin, some of which is incorporated into the cell membrane where it serves as receptor of antigen. These cells, on contact with specific antigen, proliferate and differentiate to plasma cells which synthesize and secrete large quantities of immunoglobulin. The two stages of differentiation of this cell line (generation of B-lymphocytes and antigen-driven maturation to plasma cells) are clearly separable during ontogeny and in some immune deficiency diseases. The present report describes morphologic aberrations of B-lymphocytes in two diseases in which second stage differentiation is defective.

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