On theories having a finite number of nonisomorphic countable models

1985 ◽  
Vol 50 (3) ◽  
pp. 806-808 ◽  
Author(s):  
Akito Tsuboi
Keyword(s):  
Finite Number ◽  
Positive Solutions ◽  
Linear Order ◽  
Order Property ◽  
Categorical Theory ◽  
Base Theory ◽  
Almost All ◽  
Countable Models

In this paper we shall state some interesting facts concerning non-ω-categorical theories which have only finitely many countable models. Although many examples of such theories are known, almost all of them are essentially the same in the following sense: they are obtained from ω-categorical theories, called base theories below, by adding axioms for infinitely many constant symbols. Moreover all known base theories have the (strict) order property in the sense of [6], and so they are unstable. For example, Ehrenfeucht's well-known example which has three countable models has the theory of dense linear order as its base theory.Many papers including [4] and [5] are motivated by the conjecture that every non-ω-categorical theory with a finite number of countable models has the (strict) order property, but this conjecture still remains open. (Of course there are partial positive solutions. For example, in [4], Pillay showed that if such a theory has few links (see [1]), then it has the strict order property.) In this paper we prove the instability of the base theory T0 of such a theory T rather that T itself. Our main theorem is a strengthening of the following which is also our result: if a theory T0 is stable and ω-categorical, then T0 cannot be extended to a theory T which has n countable models (1 < n < ω) by adding axioms for constant symbols.

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’.

Properties of Classes of Random Graphs

1994 ◽  
Vol 3 (4) ◽  
pp. 435-454 ◽  
Author(s):  
Neal Brand ◽  
Steve Jackson
Keyword(s):  
Random Graphs ◽  
Higher Order ◽  
Order Property ◽  
First Order ◽  
New Class ◽  
Vertex Transitive ◽  
Vertex Transitive Graphs ◽  
Almost All ◽  
Transitive Graphs

In [11] it is shown that the theory of almost all graphs is first-order complete. Furthermore, in [3] a collection of first-order axioms are given from which any first-order property or its negation can be deduced. Here we show that almost all Steinhaus graphs satisfy the axioms of almost all graphs and conclude that a first-order property is true for almost all graphs if and only if it is true for almost all Steinhaus graphs. We also show that certain classes of subgraphs of vertex transitive graphs are first-order complete. Finally, we give a new class of higher-order axioms from which it follows that large subgraphs of specified type exist in almost all graphs.

Bad models in nice neighborhoods

1986 ◽  
Vol 51 (4) ◽  
pp. 1043-1055 ◽  
Author(s):  
Terry Millar
Keyword(s):  
Linear Order ◽  
Homogeneous Model ◽  
Order Type ◽  
Saturated Model ◽  
Countable Number ◽  
Decidable Theory ◽  
Type Spectrum ◽  
Image Position ◽  
Countable Models

This paper contains an example of a decidable theory which has1) only a countable number of countable models (up to isomorphism);2) a decidable saturated model; and3) a countable homogeneous model that is not decidable.By the results in [1] and [2], this can happen if and only if the set of types realized by the homogeneous model (the type spectrum of the model) is not .If Γ and Σ are types of a theory T, define Γ ◁ Σ to mean that any model of T realizing Γ must realize Σ. In [3] a decidable theory is constructed that has only countably many countable models, only recursive types, but whose countable saturated model is not decidable. This is easy to do if the restriction on the number of countable models is lifted; the difficulty arises because the set of types must be recursively complex, and yet sufficiently related to control the number of countable models. In [3] the desired theory T is such thatis a linear order with order type ω*. Also, the set of complete types of T is not . The last feature ensures that the countable saturated model is not decidable; the first feature allows the number of countable models to be controlled.

scholarly journals On the density and the structure of the Peirce-like formulae

2008 ◽  
Vol DMTCS Proceedings vol. AI,... (Proceedings) ◽  
Author(s):  
Antoine Genitrini ◽  
Jakub Kozik ◽  
Grzegorz Matecki
Keyword(s):  
Finite Number ◽  
Upper Bound ◽  
Large Family ◽  
Partial Answer ◽  
International Audience ◽  
Almost All

International audience Within the language of propositional formulae built on implication and a finite number of variables $k$, we analyze the set of formulae which are classical tautologies but not intuitionistic (we call such formulae - Peirce's formulae). We construct the large family of so called simple Peirce's formulae, whose sequence of densities for different $k$ is asymptotically equivalent to the sequence $\frac{1}{ 2 k^2}$. We prove that the densities of the sets of remaining Peirce's formulae are asymptotically bounded from above by $\frac{c}{ k^3}$ for some constant $c \in \mathbb{R}$. The result justifies the statement that in the considered language almost all Peirce's formulae are simple. The result gives a partial answer to the question stated in the recent paper by H. Fournier, D. Gardy, A. Genitrini and M. Zaionc - although we have not proved the existence of the densities for Peirce's formulae, our result gives lower and upper bound for it (if it exists) and both bounds are asymptotically equivalent to $\frac{1}{ 2 k^2}$.

Approximation of Linear Sets in the Plane

2019 ◽  
Vol 7 (3) ◽  
pp. 60-69 ◽  
Author(s):  
В. Юрков ◽  
V. Yurkov
Keyword(s):  
Finite Number ◽  
Continuous Family ◽  
Point Sets ◽  
Multiple Points ◽  
Linear Set ◽  
Linear Sets ◽  
The Best Approximation ◽  
Special Factor ◽  
Almost All ◽  
The Given

A few general lines in the ordinary Euclidean plane are said to be line generators of a plane linear set. To be able to say that every line of the set belongs to one-parametrical line set we have to find their envelope. We thus create a pencil of lines. In this article it will be shown that there are a finite number of pencils in one linear set. To find a pencil of lines the linear parametrical approximation is applied. Almost all of problems concerning the parametrical approximation of figure sets are well known and deeply developed for any point sets. The problem of approximation for non-point sets is an actual one. The aim of this paper is to give a path to parametrical approximation of linear sets defined in plane. The sets are discrete and consist of finite number of lines without any order. Each line of the set is given as y = ax + b. Parametrical approximation means a transformation the discrete set of lines into completely continuous family of lines. There are some problems. 1. The problem of order. It is necessary to represent the chaotic set of lines as well-ordered one. The problem is solved by means of directed circuits. Any of chaotic sets has a finite number of directed circuits. To create an order means to find all directed circuits in the given set. 2. The problem of choice. In order to find the best approximation, for example, the simplest one it is necessary to choose the simplest circuit. Some criteria of the choice are discussed in the paper. 3. Interpolation the set of line factors. A direct approach would simply construct an interpolation for all line factors. But this can lead to undesirable oscillations of the line family. To eliminate the oscillations the special factor interpolation are suggested. There are linear sets having one or several multiple points, one or several multiple lines and various combinations of multiple points and lines. Some theorems applied to these cases are formulated in the paper.

On strongly minimal sets

1971 ◽  
Vol 36 (1) ◽  
pp. 79-96 ◽  
Author(s):  
J. T. Baldwin ◽  
A. H. Lachlan
Keyword(s):  
Categorical Theory ◽  
Minimal Sets ◽  
Countable Models

The purpose of this paper is twofold. In §1 and §2 which are largely expository we develop the known theory of ℵ1-categoricity in terms of strongly minimal sets. In §3 we settle affirmatively Vaught's conjecture that a complete ℵ1-categorical theory has either just one or just ℵ0 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.

Adding linear orders

2012 ◽  
Vol 77 (2) ◽  
pp. 717-725 ◽  
Author(s):  
Saharon Shelah ◽  
Pierre Simon
Keyword(s):  
Linear Order ◽  
Linear Orders ◽  
Categorical Theory

AbstractWe address the following question: Can we expand an NIP theory by adding a linear order such that the expansion is still NIP? Easily, if acl(A)=A for all A, then this is true. Otherwise, we give counterexamples. More precisely, there is a totally categorical theory for which every expansion by a linear order has IP. There is also an ω-stable NDOP theory for which every expansion by a linear order interprets pseudofinite arithmetic.

scholarly journals Chained Einstein–Podolsky–Rosen steering inequalities with improved visibility

2018 ◽  
Vol 16 (04) ◽  
pp. 1850034 ◽  
Author(s):  
Hui-Xian Meng ◽  
Jie Zhou ◽  
Changliang Ren ◽  
Hong-Yi Su ◽  
Jing-Ling Chen
Keyword(s):  
Finite Number ◽  
Threshold Values ◽  
Text Setting ◽  
Optimal Measurement ◽  
Linear Formula ◽  
Text Settings ◽  
Einstein Podolsky Rosen ◽  
Werner States ◽  
Almost All

It is known that the linear [Formula: see text]-setting steering inequalities introduced in [D. J. Saunders, S. J. Jones, H. M. Wiseman and G. J. Pryde, Nat. Phys. 6 (2010) 845.] are very efficient inequalities in detecting steerability of the Werner states by using optimal measurement axes. Here, we construct chained steering inequalities that have improved visibility for the Werner states under a finite number of settings. Specifically, the threshold values of quantum violation of our inequalities for the [Formula: see text] settings are lower than those of the linear steering inequalities. Furthermore, for almost all generalized Werner states, the chained steering inequalities always have improved visibility in comparison with the linear steering inequalities.

Decidability and ℵ0-categoricity of theories of partially ordered sets

1980 ◽  
Vol 45 (3) ◽  
pp. 585-611 ◽  
Author(s):  
James H. Schmerl
Keyword(s):  
Linear Order ◽  
Partially Ordered Set ◽  
Ordered Set ◽  
Partially Ordered Sets ◽  
Finite Width ◽  
Ordered Sets ◽  
Categorical Theory ◽  
Partially Ordered ◽  
Decidable Theory ◽  
Finite Partially Ordered Set

AbstractThis paper is primarily concerned with ℵ0-categoricity of theories of partially ordered sets. It contains some general conjectures, a collection of known results and some new theorems on ℵ0-categoricity. Among the latter are the following.Corollary 3.3. For every countable ℵ0-categoricalthere is a linear order of A such that (, <) is ℵ0-categorical.Corollary 6.7. Every ℵ0-categorical theory of a partially ordered set of finite width has a decidable theory.Theorem 7.7. Every ℵ0-categorical theory of reticles has a decidable theory.There is a section dealing just with decidability of partially ordered sets, the main result of this section beingTheorem 8.2. If (P, <) is a finite partially ordered set and KP is the class of partially ordered sets which do not embed (P, <), then Th(KP) is decidable iff KP contains only reticles.

Dimension theory and homogeneity for elementary extensions of a model

1982 ◽  
Vol 47 (1) ◽  
pp. 147-160 ◽  
Author(s):  
Anand Pillay
Keyword(s):  
Finite Number ◽  
Large Class ◽  
Countable Model ◽  
Dimension Theory ◽  
Elementary Embedding ◽  
Minimal Extension ◽  
Infinite Set ◽  
Countable Models

We take a fixed countable model M0, and we look at the structure of and number of its countable elementary extensions (up to isomorphism over M0). Assuming that S(M0) is countable, we prove that if N is a weakly minimal extension of , and if then there is an elementary embedding of N into M over M0), then N is homogeneous over M0. Moreover the condition that ∣S(M0)∣ = ℵ0 cannot be removed. Under the hypothesis that M0 contains no infinite set of tuples ordered by a formula, we prove that M0 has infinitely many countable elementary extensions up to isomorphism over M0. A preliminary result is that all types over M0 are definable, and moreover is definable over M0 if and only if is definable over M0 (forking symmetry). We also introduce a notion of relative homogeneity, and show that a large class of elementary extensions of M0 are relatively homogeneous over M0 (under the assumptions that M0 has no order and S(M0) is countable).I will now discuss the background to and motivation behind the results in this paper, and also the place of this paper relative to other conjectures and investigations. To simplify notation let T denote the complete diagram of M0. First, our result that if M0 has no order then T has infinitely many countable models is related to the following conjecture: any theory with a finite number (more than one) of countable models is unstable.

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