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Countable models of N1-categorical theories
Abstract
Countable models of ℵ1-categorical theories are classified. It is shown that such a theory has only a countable number of nonisomorphic countable
models.
... Proof. Using 2.5 we can construct in M tile first ¢o + 1 models of the elementary tower for T; i.e. a sequence ('~{,j)tza~ of countable m models of T, with ,~t 0 prime, ')in+ 1 a prime elementary extension of ~[n, and ~1~, = LI ~1 n (see [9] or [ 1 ]: ~,~1,~+ 1 is a prirae model of Th((~ln+l)nnu(b}) n<to for a suitable b ~ An÷ 1 ). Morley's result [9] shows that any countable model of T is isomorphic to one from this sequence. ...
... Using 2.5 we can construct in M tile first ¢o + 1 models of the elementary tower for T; i.e. a sequence ('~{,j)tza~ of countable m models of T, with ,~t 0 prime, ')in+ 1 a prime elementary extension of ~[n, and ~1~, = LI ~1 n (see [9] or [ 1 ]: ~,~1,~+ 1 is a prirae model of Th((~ln+l)nnu(b}) n<to for a suitable b ~ An÷ 1 ). Morley's result [9] shows that any countable model of T is isomorphic to one from this sequence. ...
... Proof. We will pt~ove that any countable model of T b as a prime elementary extension; by the result of Morley [9] this is sufficient for T to be l -categorical. First we show that any couiatabl~ t model of T (necessarily in M) has a prime elementary extension. ...
... A lot of specialists tried to solve this problem and received partially negative solutions proving for various subclasses of the class of stable theories that theories from these subclasses have one or infinitely many countable models up to isomorphism. Among stable theories having such number of countable models we would mention the followings: uncountably categorical theories (Morley [4], Baldwin -Lachlan [5]), superstable theories (Lachlan [6]), normal and weakly normal theories (Pillay [7,8]), unions of increasing chains of pseudosuperstable theories (Tsuboi [9]), theories admitting finite codings (Hrushovski [10]) . From the other hand the problem is still open and there is a list of properties that any stable theory with a finite (> 1) number of countable models must have. ...
... , c < πr, 0 < α, β, γ < π, a, b, c and α, β, γ are sides and angles of some triangle on S ∪∆ 0 (G 1 , G 2 ) . Obviously S is a trigonometrical set, and if in its connected trigonometry we factorize the group G 1 by the group {kπr | k ∈ Z}, + , identify correspondent points on lines, symmetrize lengthes and angle values, identify lines having the angles ±π and their correspondent points as above, we obtain the spherical trigonometry on S. 4. Consider an interpretation of the Poincaré's model of the Lobachevskian plane P L [18] as a trigonometry, in which the points are the inner points of some circle K, and the lines are the parts of circles lying in K and intersecting the border of K under the right angle. ...
... a n α 1 . . . α n ∈ S(pm) iff α 1 [1,2], [1,4], [1,6], [2,3], [2,5], [3,4], [3,6], [4,5], [5,6] ...
At this survey we present the motivation and the main results concerning the notions of group polygonometries and the algebraic systems connected with them. We shall use some standard terminology of the model theory [1], the general algebra [2] and the graph theory [3]. 1. Some historical remarks. In the early of seventies the problem of the existence of a stable complete first order theory having a finite (> 1) number of countable models has arised. A lot of specialists tried to solve this problem and received partially negative solutions proving for various sub-classes of the class of stable theories that theories from these subclasses have one or infinitely many countable models up to isomorphism. Among sta-ble theories having such number of countable models we would mention the followings: uncountably categorical theories (Morley [4], Baldwin — Lach-lan [5]), superstable theories (Lachlan [6]), normal and weakly normal theo-ries (Pillay [7, 8]), unions of increasing chains of pseudosuperstable theories (Tsuboi [9]), theories admitting finite codings (Hrushovski [10]). From the other hand the problem is still open and there is a list of prop-erties that any stable theory with a finite (> 1) number of countable models must have. Among them we mention the existence of a nonisolated powerful type (Benda [11]), i.e. of a type p(¯ x) ∈ S(∅) such that any model realizing p(¯ x) has realizations of any type over ∅. From the compactness theorem and [7, 12] we get the following FACT 1.1. If p(¯ x) is a nonisolated powerful type, then for any formula ϕ(¯ x) ∈ p(¯ x) there is a formula ψ(¯ x, ¯ y), l(¯ x) = l(¯ y) such that 1) for any ¯ a, ¯ b |= p there is ¯ c satisfying ϕ(¯ x) such that |= ψ(¯ a, ¯ c) ∧ ψ(¯ b, ¯ c); 2) the directed graph on the set P of realizations of p(¯ x) with the relation {(¯ a, ¯ b) | |= ψ(¯ a, ¯ b)} doesn't have contours; 1 The research supported by the Russian foundation of fundamental researches (96-01-01675).
... In [3] l^os conjectured that in analogy with the theory of algebraically closed fields of fixed characteristic or with the theory of torsion-free divisible abelian groups, a theory is categorical in one uncountable power just if it is categorical in all uncountable powers. M. Morley [4] proved the Los conjecture and himself conjectured that for every countable T 9 n(T, X) is a non-decreasing function of X with one exception: those T 9 like the theory of algebraically closed fields of fixed characteristic, where n(T 9 H 0 ) = N 0 and n(T 9 X) = 1 for uncountable X. This exceptional case was completely investigated by Morley [4], [5] who showed that if n(T 9 K) = 1 for some uncountable K then n(T 9 K) = 1 for all uncountable K and n(T 9 N 0 ) < N 0 and Baldwin and Lachlan [1] who showed n(T 9 N x ) = 1 implies n(T 9 KQ) is 1 or N 0 . ...
... M. Morley [4] proved the Los conjecture and himself conjectured that for every countable T 9 n(T, X) is a non-decreasing function of X with one exception: those T 9 like the theory of algebraically closed fields of fixed characteristic, where n(T 9 H 0 ) = N 0 and n(T 9 X) = 1 for uncountable X. This exceptional case was completely investigated by Morley [4], [5] who showed that if n(T 9 K) = 1 for some uncountable K then n(T 9 K) = 1 for all uncountable K and n(T 9 N 0 ) < N 0 and Baldwin and Lachlan [1] who showed n(T 9 N x ) = 1 implies n(T 9 KQ) is 1 or N 0 . ...
... One motivation for studying the SB-property is that if T has this property, then T would be a theory for which we have a "good understanding" of its models, in terms that they are classified by some reasonable collection of invariants. For example, by Morley's theorem, see [28], if T is countable and ℵ 1 -categorical then the models of T are classified by a single invariant cardinal number that is preserved by elementary embeddings, so T would have SBproperty; however SB-property is a weaker condition than ℵ 1 -categoricity, for example the theory of an infinite set with a predicate which is infinite and coinfinite has the SB-property and it is not ℵ 1 -categorical. The SB-property for first order theories have been well studied in [20], [21] and [29]. ...
A complete theory T has the Schr\"oder-Bernstein property or simply the SB-property if any pair of elementarily bi-embeddable models are isomorphic. This property has been studied in the discrete first-order setting and can be seen as a first step towards classification theory. This paper deals with the SB-property on continuous theories. Examples of complete continuous theories that have this property include Hilbert spaces and probability algebras with or without atoms. We also study a weaker notion, the SB-property up to perturbations. This property holds if any two bi-embeddable models are isomorphic up to perturbations. We prove that the theory of Hilbert spaces expanded with a bounded self-adjoint operator has the SB-property up to perturbations of the operator and that the theory of atomless probability algebras with a generic automorphism have the SB-property up to perturbations of the automorphism. Finally we study how the SB-property behaves with respect to randomizations.
... The first two concern ℵ 1 but not ℵ 0 -categorical theories. Morley partially answers the first in [20] proving that the countable models can be arranged in a chain of length at most . Baldwin and Lachlan [2] showed non-trivial finite chains are impossible. ...
... Let T be a countable finitary theory that is to~-categorical. Morley [14] showed that T has at most countably many countable models and that any two countable models of T are comparable in the sense of elementary embedding. Marsh [13] proved that T has a finite inessential expansion T* such that T* has either one or countably many countable models. ...
A complete theory T has the Schröder–Bernstein property or simply the SB-property if any pair of elementarily bi-embeddable models are isomorphic. This property has been studied in the discrete first-order setting and can be seen as a first step towards classification theory. This paper deals with the SB-property on continuous theories. Examples of complete continuous theories that have this property include Hilbert spaces and any completion of the theory of probability algebras. We also study a weaker notion, the SB-property up to perturbations. This property holds if any two elementarily bi-embeddable models are isomorphic up to perturbations. We prove that the theory of Hilbert spaces expanded with a bounded self-adjoint operator has the SB-property up to perturbations of the operator and that the theory of atomless probability algebras with a generic automorphism have the SB-property up to perturbations of the automorphism. We also study how the SB-property behaves with respect to randomizations. Finally we prove, in the continuous setting, that if T is a strictly stable theory then T does not have the SB-property.
This chapter emphasizes that many superficially unrelated results of model theory are consequences of the same simple theorems on omitting types, discusses several properties of omitting types, and also illustrates their uses with a variety of example. By systematically developing certain rather elementary observations on omitting types, many known results can be known and unified, a number of new ones can be found, and some classical results can be trivialized. The chapter discusses Lindstrom's theorem showing that the model completeness is a consequence of a very simple observation on omitting types in theories that are categorical in some infinite power. The chapter also discusses The Completeness Theorem, the downward Löwenheim–Skolem theorem, existentially closed models, the amalgamation property, the omitting-types theorem, other theorems, and lemmas.
The paper studies the history and recent developments in non-elementary model theory focusing in the framework of ab-stract elementary classes. We discuss the role of syntax and seman-tics and the motivation to generalize first order model theory to non-elementary frameworks and illuminate the study with concrete examples of classes of models. This first part introduces the main conceps and philosophies and discusses two research questions, namely categoricity transfer and the stability classification.
We investigate in detail stable formulas, ranks of types and their definability, the f.c.p., some syntactical properties of unstable formulas, indiscernible sets and degrees of types in superstable theories. There is a last of all results connected with those properties, or whose proof use them.
Then i) ~ is a universal enumerated set; 2) for arbitrary subsets ~O,,«,~~~ such that a) d~ ~~ ~~ for &~~,d~/
If ~ is an algebraic system we then denote by A its basic set. If ~ is an L -system, X ~ A, then the language L (X) is obtained by adding to L the set of constants {C a r ~ ~ X }. We denote by (~, X ) the natural enrichment of ~ to an L (X) -system. F n (L) denotes the set of formulas of language L all of whose free variables enter into ~o, "', z)n-¢} • ~ CX) is a contraction of F~ (L CX)), while the symbol S~(×) denotes the set of zv-types of theory Y/t(~,X). If Ca F CZ) thenthe symbol ~(~J denotes the set {ae A I cZ ~ ~Co)} - The symbol IXl denotes the cardinality (power) of set X .
Since their inception, the Perspectives in Logic and Lecture Notes in Logic series have published seminal works by leading logicians. Many of the original books in the series have been unavailable for years, but they are now in print once again. In this volume, the twelfth publication in the Perspectives in Logic series, John T. Baldwin presents an introduction to first order stability theory, organized around the spectrum problem: calculate the number of models a first order theory T has in each uncountable cardinal. The author first lays the groundwork and then moves on to three sections: independence, dependence and prime models, and local dimension theory. The final section returns to the spectrum problem, presenting complete proofs of the Vaught conjecture for ω-stable theories for the first time in book form. The book provides much-needed examples, and emphasizes the connections between abstract stability theory and module theory.
We study the notion of definable type, and use it to define theproduct of types and theheir of a type. Then, in the case of stable and superstable theories, we make a general study of the notion of rank. Finally,
we use these techniques to give a new proof of a theorem of Lachlan on the number of isomorphism types of countable models
of a superstable theory.
This paper deals with the notion of prime extension defined by Morley. A connection is established between the categoricity
of a theory in a cardinal greater than the power of its language and the existence of prime extensions. For such theories
we prove a minimality property of the prime extensions and we conclude by a property of the group of automorphisms of their
models.
It is shown that, For each complete theoryT, the nomberh
T(m) of homogeneous models ofT of powerm is a non-increasing function of uncountabel cardinalsm Moreover, ifh
T(ℵ0)≦ℵ0, then the functionh
T is also non-increasing ℵ0 to ℵ1.
In this article we define when a finite diagram of a model is stable, we investigate what is the form of the class of powers in which a finite diagram is stable, and we generalize some properties of totally transcendental theories to stable finite diagrams. Using these results we investigate several theories which have only homogeneous models in certain power. We also investigate when there exist models of a certain diagram which are λ-homogenous and not λ+-homogeneous in various powers. We also have new results about stable theories and the existence of maximally λ-saturated models of power μ.
A first-order theory T has the Schr\"oder-Bernstein (SB) property if any pair
of elementarily bi-embeddable models are isomorphic. We prove that T has an
expansion by constants that has the SB property if and only if T is superstable
and non-multidimensional. We also prove that among superstable theories T, the
class of a-saturated models of T has the SB property if and only if T has no
nomadic types.
The purpose of this paper is twofold. In §1 and §2 which are largely expository we develop the known theory of ℵ 1 -categoricity in terms of strongly minimal sets. In §3 we settle affirmatively Vaught's conjecture that a complete ℵ 1 -categorical theory has either just one or just ℵ 0 countable models, and in §4 we present an example which serves to illustrate the ideas of §3.
As far as we know the only work published on strongly minimal sets is that of Marsh [3]. The present exposition goes beyond [3] in showing that any ℵ-categorical theory has a principal extension in which some formula is strongly minimal.
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.
In [3] we have associated to a structure an ordinal which gives us information about elementary substructures of the structure. For example a structure whose ascending chain number (as we call the ordinal) is ω could be called Noetherian since all ascending elementary chains inside it are finite (and there are arbitrarily large finite chains). Theorem 2 shows that such structures exist. In fact we prove that for any α < ω1 there is a structure whose ascending chain number is α. The construction is based on the existence of a certain group of permutations of ω (see Theorem 1). The second part of this paper deals with the relevance of the chain number to the study of Jonsson algebras.
Meeting of the Association for Symbolic Logic, Atlanta 1973 - Volume 39 Issue 2 - C. Ward Henson, Bjarni Jónsson, E. G. K. Lopez-Escobar, Michael D. Resnik
If T is a complete first-order totally transcendental theory then over every T-structure A there is a prime model unique up to isomorphism over A. Moreover M is a prime model over A iff: (1) every finite sequence from M realizes an isolated type over A, and (2) there is no uncountable indiscernible set over A in M.
The existence of prime models was proved by Morley [3] and their uniqueness for countable A by Vaught [9]. Sacks asked (see Chang and Keisler [1, question 25]) whether the prime model is unique. After proving this I heard Ressayre had proved that every two strictly prime models over any T -structure A are isomorphic, by a strikingly simple proof. From this follows
If T is totally transcendental, M a strictly prime model over A then every elementary permutation of A can be extended to an automorphism of M. (The existence of M follows by [3].)
By our results this holds for any prime model. On the other hand Ressayre's result applies to more theories. For more information see [6, §0A]. A conclusion of our theorem is the uniqueness of the prime differentially closed field over a differential field. See Blum [8] for the total transcendency of the theory of differentially closed fields.
We can note that the prime model M over A is minimal over A iff in M there is no indiscernible set over A (which is infinite).
On the categoricity in power of elementary deductive theories
- J Łos
On ω 1-categorical but not ω-categorical theories, Doctoral dissertation
- W Marsh
W. Marsh, On O~rcategorical but not co-categorical theories, Doctoral dissertation, Dartmouth College, June, 1966.
A condition equivalent to Nl-categoricity, Notices Amer
- M Morley
M. Morley, A condition equivalent to Nl-categoricity, Notices Amer. Math. Soc. 11 (1964), 687.
On theories categorical in power ≦ ℵ 0
- C Ryll-Nardzewski
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A condition equivalent to ℵ 1-categoricity
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