scholarly journals Generalized prime models

1971 ◽  
Vol 36 (4) ◽  
pp. 593-606 ◽  
Author(s):  
Robert Fittler
Keyword(s):  
Order Theory ◽  
Complete Theory ◽  
Prime Model ◽  
First Order ◽  
First Order Theory ◽  
General Conditions

A prime model O of some complete theory T is a model which can be elementarily imbedded into any model of T (cf. Vaught [7, Introduction]). We are going to replace the assumption that T is complete and that the maps between the models of T are elementary imbeddings (elementary extensions) by more general conditions. T will always be a first order theory with identity and may have function symbols. The language L(T) of T will be denumerable. The maps between models will be so called F-maps, i.e. maps which preserve a certain set F of formulas of L(T) (cf. I.1, 2). Roughly speaking a generalized prime model of T is a denumerable model O which permits an F-map O→M into any model M of T. Furthermore O has to be “generated” by formulas which belong to a certain subset G of F.

An Example of a Complete First-Order Theory with All Models Algorithmically Trivial but Without Locally Finite Models

1982 ◽  
Vol 5 (3-4) ◽  
pp. 313-318
Author(s):  
Paweł Urzyczyn
Keyword(s):  
Order Theory ◽  
Complete Theory ◽  
Prime Model ◽  
Program Schema ◽  
Locally Finite ◽  
First Order ◽  
Finite Models ◽  
First Order Theory

We show an example of a first-order complete theory T, with no locally finite models and such that every program schema, total over a model of T, is strongly equivalent in that model to a loop-free schema. For this purpose we consider the notion of an algorithmically prime model, what enables us to formulate an analogue to Ryll-Nardzewski Theorem.

Nonarithmetical ℵ0-categorical theories with recursive models

1994 ◽  
Vol 59 (1) ◽  
pp. 106-112 ◽  
Author(s):  
Julia F. Knight
Keyword(s):  
Order Theory ◽  
Complete Theory ◽  
First Order ◽  
Recursive Model ◽  
Recursive Models ◽  
First Order Theory ◽  
Free Variables

In what follows, L is a recursive language. The structures to be considered are L-structures with universe named by constants from ω. A structure is recursive A if the open diagram D() is recursive. Lerman and Schmerl [L-S] proved the following result.Let T be an ℵ0-categorical elementary first-order theory. Suppose that for all n, , and T is arithmetical. Then T has a recursive model.The aim of this paper is to extend Theorem 0.1. Stating the extension requires some terminology. Consider finitary formulas with symbols from L and sometimes extra constants from ω. For each n ∈ ω, the Σn and Πn formulas are as usual. Then Bnformulas are Boolean combinations of Σn formulas. For an L-structure , Dn() denotes the set of Bn sentences in the complete diagram Dc(). A complete Σn theory is a maximal consistent set of ΣnL-sentences. We may write φ(x), or Γ(x), to indicate that the free variables of the formula φ, or the set Γ, are among those in x. A complete Bn type for x is a maximal consistent set Γ(x) of Bn formulas with just the free variables x.If T is ℵ0-categorical, then for each x only finitely many complete types Γ(x) are consistent with T. While Lerman and Schmerl stated their result just for ℵ0-categorical theories, essentially the same proof yields the following.Theorem 0.2. Let T be a consistent, complete theory such that for all n andx, only finitely many complete Bn types Γ(x) are consistent with T.

Prime e.c. commutative rings in characteristic n ≥ 2

1999 ◽  
Vol 64 (2) ◽  
pp. 629-633
Author(s):  
Dan Saracino
Keyword(s):  
Commutative Rings ◽  
Order Theory ◽  
Prime Model ◽  
Finite Model ◽  
First Order ◽  
First Order Theory ◽  
Building Models ◽  
Open Questions ◽  
A Chain ◽  
Mathematics Department

Let CR denote the first-order theory of commutative rings with unity, formulated in the language L = 〈 +, •, 0, 1〉. Virtually everything that is known about existentially complete (e.c.) models of CR is contained in Cherlin's paper [2], where it is shown, in particular, that the e.c. models are not first-order axiomatizable. The purpose of this note is to show that, in analogy with the case of fields, there exists a unique prime e.c. model of CR in each characteristic n > 2. As a consequence we settle Problem 8 in the list of open questions at the end of Hodges' book Building models by games ([5], p. 278).By a “prime” e.c. model of characteristic n ≥ 2 we mean one that embeds in every e.c. model of characteristic n. (The embedding is not always elementary, since [2] not all e.c. models of characteristic n are elementarily equivalent.) The prime model is characterized by the fact that it is the union of a chain of finite subrings each of which is an amalgamation base for CR. In §1 we describe the finite amalgamation bases for CR and show that every finite model embeds in a finite amalgamation base; in §2 we use this information to obtain prime e.c. models and answer Hodges' question.Our results on prime e.c. models were obtained some years ago, during the fall term of 1982, while the author was a visitor at Wesleyan University. The author wishes to take this opportunity to thank the mathematics department at Wesleyan for its hospitality during that visit, and subsequent ones.

Models with second order properties in successors of singulars

1989 ◽  
Vol 54 (1) ◽  
pp. 122-137
Author(s):  
Rami Grossberg
Keyword(s):  
Order Theory ◽  
Second Order ◽  
Order Logic ◽  
First Order Logic ◽  
Complete Theory ◽  
Singular Cardinal ◽  
Strongly Compact Cardinal ◽  
Compact Cardinal ◽  
First Order ◽  
First Order Theory

AbstractLet L(Q) be first order logic with Keisler's quantifier, in the λ+ interpretation (= the satisfaction is defined as follows: M ⊨ (Qx)φ(x) means there are λ+ many elements in M satisfying the formula φ(x)).Theorem 1. Let λ be a singular cardinal; assume □λ and GCH. If T is a complete theory in L(Q) of cardinality at most λ, and p is an L(Q) 1-type so that T strongly omits p( = p has no support, to be defined in §1), then T has a model of cardinality λ+ in the λ+ interpretation which omits p.Theorem 2. Let λ be a singular cardinal, and let T be a complete first order theory of cardinality λ at most. Assume □λ and GCH. If Γ is a smallness notion then T has a model of cardinality λ+ such that a formula φ(x) is realized by λ+ elements of M iff φ(x) is not Γ-small. The theorem is proved also when λ is regular assuming λ = λ<λ. It is new when λ is singular or when ∣T∣ = λ is regular.Theorem 3. Let λ be singular. If Con(ZFC + GCH + ∃κ) [κ is a strongly compact cardinal]), then the following is consistent: ZFC + GCH + the conclusions of all above theorems are false.

Prime models and almost decidability

1986 ◽  
Vol 51 (2) ◽  
pp. 412-420 ◽  
Author(s):  
Terrence Millar
Keyword(s):  
Order Theory ◽  
The Other ◽  
Prime Model ◽  
First Order ◽  
Decidable Theory ◽  
Complete Extension ◽  
First Order Theory ◽  
Additional Constant ◽  
Open Question ◽  
Countable Models

This paper introduces and investigates a notion that approximates decidability with respect to countable structures. The paper demonstrates that there exists a decidable first order theory with a prime model that is not almost decidable. On the other hand it is proved that if a decidable complete first order theory has only countably many complete types, then it has a prime model that is almost decidable. It is not true that every decidable complete theory with only countably many complete types has a decidable prime model. It is not known whether a complete decidable theory with only countably many countable models up to isomorphism must have a decidable prime model. In [1] a weaker result was proven—if every complete extension, in finitely many additional constant symbols, of a theory T fails to have a decidable prime model, then T has 2ω nonisomorphic countable models. The corresponding statement for saturated models is false, even if all the complete types are recursive, as was shown in [2]. This paper investigates a variation of the open question via a different notion of effectiveness—almost decidable.A tree Tr will be a subset of ω<ω that is closed under predecessor. For elements f, g in ω<ω ∪ ωω, ƒ ⊲ g iffdf ∀i < lh(ƒ)[ƒ(i) = g(i)].

Prime models of finite computable dimension

2009 ◽  
Vol 74 (1) ◽  
pp. 336-348
Author(s):  
Pavel Semukhin
Keyword(s):  
Order Theory ◽  
Prime Model ◽  
Computable Model ◽  
Undirected Graphs ◽  
Integral Domains ◽  
Computable Model Theory ◽  
First Order ◽  
First Order Theory ◽  
Computable Dimension ◽  
Open Question

AbstractWe study the following open question in computable model theory: does there exist a structure of computable dimension two which is the prime model of its first-order theory? We construct an example of such a structure by coding a certain family of c.e. sets with exactly two one-to-one computable enumerations into a directed graph. We also show that there are examples of such structures in the classes of undirected graphs, partial orders, lattices, and integral domains.

SEPARABLE MODELS OF RANDOMIZATIONS

2015 ◽  
Vol 80 (4) ◽  
pp. 1149-1181 ◽  
Author(s):  
URI ANDREWS ◽  
H. JEROME KEISLER
Keyword(s):  
Probability Density Function ◽  
Probability Density ◽  
Density Function ◽  
Order Theory ◽  
Prime Model ◽  
First Order ◽  
First Order Theory ◽  
Separable Models ◽  
Homogeneous Models ◽  
Countable Models

AbstractEvery 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 TR 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.

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