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On the number of homogeneous models of a given power
Abstract
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.
... Shelah solved this problem by listing the possible spectrum functions I T (κ), the number of models of T of cardinality κ, and noting that all were non-decreasing on uncountable cardinals. Morley and Keisler [11] made progress on this question for homogenous models under GCH. Full clarification appeared in [10,31,32]. ...
... Section 6 in [8]). By Lemma 6.3 in [8], if μ → (ν) 2 2 then every totally ordered set of cardinality μ contains a ν-sequence. By the Erd˝os-Rado theorem, we have ...
We obtain some results on existence of small extensions of models of weakly o-minimal atomic theories. In particular, we find a sharp upper estimate for the Hanf number of such a theory for omitting
an arbitrary family of pure types. We also find a sharp upper estimate for cardinalities of weakly o-minimal absolutely homogeneous models and a sufficient condition for absolute homogeneity.
... By induction we define an increasing sequence of types pi SUCl~ that p0 = p and p~ ~ SD(AZ) @BULLET for i = 6 a limit ordinal, take pt = O pi, and ifp i is j<z defined we let p~+l be an extension ofp z in SD(A~ u {a~} ) = SD(A z+l ) (as guaranteed by the previous paragraph), p IA I is the required type. Remark: This partially selves a problem from Keisler and Morley [5]. Proof: We fir.,,t prove that every (D, g)-homogeneous model M is of power >/s. ...
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.
... The most pleasing and useful property of homogeneous models and of the set of pure types T(A) is the following Uniqueness Theorem for Homogeneous Models. Theorem 3.3 (Morley and Keisler [30]). Given a countable complete theory T and homogeneous models A and B of T of the same cardinality, T(A) = T(B) =⇒ A ∼ = B. ...
A Turing degree d is homogeneous bounding if every complete decidable (CD) theory has a d-decidable homogeneous model 풜, i.e., the elementary diagram De(풜) has degree d. It follows from results of Macintyre and Marker that every PA degree (i.e., every degree of a complete extension of Peano Arithmetic) is homogeneous bounding. We prove that in fact a degree is homogeneous bounding if and only if it is a PA degree. We do this by showing that there is a single CD theory T such that every homogeneous model of T has a PA degree.
... The main theorem 2.8, the existence and uniqueness of the model homogeneous models, is from Jonsson [Jo56], [Jo60]. The result on the number of D-homogeneous universal models is from Keisler and Morley [KM67]. Probably a new result is 2.5(2) (and 2.17). ...
We prove in ZFC, no psi in L_{omega_1,omega}[Q] have unique model of uncountable cardinality, this confirms theBaldwin conjecture. But we analyze this in more general terms. We introduce and investigate a.e.c. and also versions of limit models, and prove some basic properties like representation by PC class, for any a.e.c. For PC_{aleph_0}-representable a.e.c. we investigate the conclusion of having not too many non-isomorphic models in aleph_1 and aleph_2, but have to assume 2^{aleph_0}<2^{aleph_1} and even 2^{aleph_1}<2^{aleph_2}.
In this paper we study the notion of a D -saturated model, which occupies an intermediate position between the notions of a homogeneous model and a saturated model. Both the homogeneous models and the saturated models play a very important role in model theory. For example, the saturated models are the universal domains of the corresponding theories in the sense that is used in algebraic geometry (one can also notice that the algebraically closed fields of infinite transcendence degree, which are the universal domains in algebraic geometry, are the saturated models of the theory of algebraically closed fields); this means that the saturated models realize all what is necessary for the effective study of the corresponding theory.The D -saturated models proved to be useful in various situations. Naturally the question arises on the conditions under which a model is D -saturated. In this paper we indicate some conditions of such sort for weakly o-minimal models and models of stable theories. Namely, for such models we prove that homogeneity and a certain approximation of D saturation imply D -saturation.
Every complete first order theory has a corresponding complete theory in continuous logic, called the randomization theory. It has two sorts, a sort for random elements of models of the first order theory, and a sort for events. In this paper we establish connections between properties of countable models of a first order theory and corresponding properties of separable models of the randomization theory. We show that the randomization theory has a prime model if and only if the first order theory has a prime model. And the randomization theory has the same number of separable homogeneous models as the first order theory has countable homogeneous models. We also show that when T has at most countably many countable models, each separable model of T
R
is uniquely characterized by a probability density function on the set of isomorphism types of countable models of T. This yields an analogue for randomizations of the results of Baldwin and Lachlan on countable models of ω1-categorical first order theories.
We obtain results on existence of dense free subgroups of the automorphism group of a homogeneous model.
We introduce the notion of a superstructure over a model. This is a generalization of the notion of the hereditarily finite superstructure ℍ\(
\mathbb{F}\mathfrak{M}
\) over a model \(
\mathfrak{M}
\). We consider the question on cardinalities of definable (interpretable) sets in superstructures over λ-homogeneous and λ-saturated models.
Some properties concerning homogeneity and the existence of small extensions proved earlier for models of superstable theories, are proved here for models of the theory of linear ordering.
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. This volume, the eighth publication in the Perspectives in Logic series, brings together several directions of work in model theory between the late 1950s and early 1980s. It contains expository papers by pre-eminent researchers. Part I provides an introduction to the subject as a whole, as well as to the basic theory and examples. The rest of the book addresses finitary languages with additional quantifiers, infinitary languages, second-order logic, logics of topology and analysis, and advanced topics in abstract model theory. Many chapters can be read independently.
Every complete first order theory has a corresponding complete theory in continuous logic, called the randomization theory. It has two sorts, a sort for random elements of models of the first order theory, and a sort for events. In this paper we establish connections between properties of countable models of a first order theory and corresponding properties of separable models of the randomization theory. We show that the randomization theory has a prime model if and only if the first order theory has a prime model. And the randomization theory has the same number of separable homogeneous models as the first order theory has countable homogeneous models.
The randomization of a complete first order theory T is the complete con-tinuous theory T R with two sorts, a sort for random elements of models of T , and a sort for events in an underlying probability space. We show what the separable models of T R look like when T has at most countably many countable models. In that case, each separable model of T R is uniquely characterized by a probability density function on the set of isomorphism types of countable models of T .
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.
A stable theory is called unidimensional if any two of its nonalgebraic types are nonorthogonal. In the paper we consider some problems concerning homogeneous models of unidimensional theories. For homogeneous models, in particular, we prove an analog of the theorem of Shelah which states that the concepts of unidimensionality and saturation are equivalent in any sufficiently saturated model of that theory.
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.
Bjarni Jdnsson's publications in algebra have had a major impact on the modern study of algebraic systems. The papers represent a number of themes, to which J6nsson has often returned over the years. These themes and the resulting papers can be grouped informally as follows. Boolean algebras and Boolean algebras with operators, product decompositions of algebraic systems, other constructions in a general setting, other constructions specifically for varieties, congruence-distributive varieties, applications to groups and semigroups, lattices as examples of algebras, and expository works. This paper will be in two parts: in Section 2, a guide to Jtnsson's contributions in many of these areas, and in Section 3, a more detailed summary of his work on congruence distributivity and the developments to which it has led. The work in lattice theory is not included here. A bibliography of a number of papers referencing Jtnsson's work in algebra is provided as an indication of his influence. The standard introductory material in universal algebra will be assumed. The notation and terminology used will not necessarily be that of the sources. Recall than an algebra A is said to be congruence-distributive if its lattice of congruence relations is distributive, congruence-modular if its lattice of congruence relations is modular, and congruence-permutable if for any two congruence relations 0, q~ the relational compositions 0 o ~b and ~b o 0 coincide. These adjectives are applied to a variety if all of its members have the property. Recall also that congruence distributivity and congruence permutability each imply congruence modularity. Following Jdnsson's usage, lattice operations will be called sums and products. A bibliographical reference such as [J3] refers to Jtnsson's publication list; [3] refers to item 3 in the bibliography. General references are [J40], [63], [74], [173], [282], and [384]. Other surveys of portions of the areas covered by this paper will be mentioned below.
Some fragment is studied of stability theory in the category of D-sets. Conditions are given for existence of D-homogeneous models of however large power. A categoricity theorem is proven for the class of (D,)-homogeneous models.
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.
Conditions are found under which homogeneity of a model M implies homogeneity of the modelM
eq. One of these conditions is ω-stability of the theory of M. Examples are constructed of superstable theories for which homogeneity is not preserved under such a passage.
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 ℵ 1 nonisomorphic countable models, then it has such models. He conjectures that if a countable theory T has more than ℵ 0 nonisomorphic countable models, then it has such models. In this paper we show that if a countable theory T has more than ℵ 0 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 ℵ 0 types, then T has nonisomorphic countable homogeneous models .
Proof. Suppose that T has more than ℵ 0 types. From [2, Corollary 2.4] T has types. Let σ be a Ttype with n variables, and T ′ = T ⋃ { σ ( c 1 , …, c n )}, where c 1 , …, c n are new constants. T ′ is consistent and has a countable model ( , a 1 , …, a n ). 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 ℵ 0 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 L α ( 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 = L 0 is countable.
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.
In the last five years there have been a number of results about the computable content of the prime, saturated, or homogeneous models of a complete decidable (CD) theory T in the spirit of Vaught's "Denumerable models of complete theories" combined with computability methods for (Turing) degrees d ≤ 0′. First we recast older results by Goncharov, Peretyat'kin, and Millar in a more modern framework which we then apply. Then we survey recent results by Lange, "The degree spectra of homogeneous models," which generalize the older results and which include positive results on when a certain homogeneous model of T has an isomorphic copy of a given Turing degree. We then survey Lange's "A characterization of the 0-basis homogeneous bounding degrees" for negative results about when does not have such copies, generalizing negative results by Goncharov, Peretyat'kin, and Millar. Finally, we explain recent results by Csima, Harizanov, Hirschfeldt, and Soare, "Bounding homogeneous models," about degrees d that are homogeneous bounding and explain their relation to the PA degrees (the degrees of complete extensions of Peano Arithmetic).
On the one hand we try to understand complete types over somewhat saturated
model of a complete first order theory which is dependent, by "decomposition
theorems for such types". Our thesis is that the picture of dependent theory is
the combination of the one for stable theories and the one for the theory of
dense linear order or trees (and first we should try to understand the quite
saturated case). On the other hand as a measure of our progress, we give
several applications considering some test questions; in particular we try to
prove the generic pair conjecture and do it for measurable cardinals. The order
of the sections is by their conceptions, so there are some repetitions.
Countable models of ℵ1-categorical theories are classified. It is shown that such a theory has only a countable number of nonisomorphic countable
models.
The θ-completeness theorem is applied to prove theorems above two-cardinal models, homogeneous models, and categoricity in
power in θ-logic.
Then the union is homogeneous of power p. This completes the proof. Theorem B is now immediate. By Lemma 6, for each homogeneous model 9~ of T of power 1t there is a homogeneous model ~3 of power m with S(~3) = S(~) By Lemma 3 we have ha(m) > ha(11). Corollary C follows from A
- On The Number Of Homogeneous Models Of A Given Power
ON THE NUMBER OF HOMOGENEOUS MODELS OF A GIVEN POWER 77 regular m, N1 _--< m < p, with S(9~.T)= S(9~. Then the union is homogeneous of power p. This completes the proof. Theorem B is now immediate. By Lemma 6, for each homogeneous model 9~ of T of power 1t there is a homogeneous model ~3 of power m with S(~3) = S(~). By Lemma 3 we have ha(m) > ha(11). Corollary C follows from A, B, and Lemma 4. Our lemmas give more general versions of Theorems A, B, and Corollary C.
o be the logic which is like first order logic but has countably infinite conjunctions, and let L~,o,l be the logic which has countably infinite conjunctions and quantifiers over countable sequences of variables
- L Let
Let L,o,,o be the logic which is like first order logic but has countably infinite conjunctions, and let L~,,o,l be the logic which has countably infinite conjunctions and quantifiers over countable sequences of variables (cf.
Some fundamental problems concerning languages with infinitely long expressions, Doctoral dissertation
- W Hanf