Finite extensions and the number of countable models

An Ehrenfeucht theory is a complete first order theory with exactly n countable models up to isomorphism, 1 < n < ω. Numerous results have emerged regarding these theories ([1]–[15]). A general question in model theory is whether or not the number of countable models of a complete theory can be different than the number of countable models of a complete consistent extension of the theory by finitely many constant symbols. Examples are known of Ehrenfeucht theories that have complete extensions by finitely many constant symbols such that the extensions fail to be Ehrenfeucht ([4], [8], [13]). These examples are easily modified to allow finite increases in the number of countable models.This paper contains examples in the other direction—complete theories that have consistent extensions by finitely many constant symbols such that the extensions have fewer countable models. This answers affirmatively a question raised by, among others, Peretyat'kin [8]. The first example will be an Ehrenfeucht theory with exactly four countable models with an extension by a constant symbol that has only three countable models. The second example will be a complete theory that is not Ehrenfeucht, but which has an extension by a constant symbol that is Ehrenfeucht. The notational conventions for this paper are standard.Peretyat'kin introduced the theory of a dense binary branching tree with a meet operator [7]. Dense ω-branching trees have also proven useful [5], [11]. Both of the Theories that will be constructed make use of dense ω-branching trees.

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Countable models of trivial theories which admit finite coding

Journal of Symbolic Logic ◽

10.2307/2275816 ◽

1996 ◽

Vol 61 (4) ◽

pp. 1279-1286 ◽

Cited By ~ 1

Author(s):

James Loveys ◽

Predrag Tanović

Keyword(s):

Order Theory ◽

First Order ◽

First Order Theory ◽

Superstable Theories ◽

Countable Models

AbstractWe prove:Theorem. A complete first order theory in a countable language which is strictly stable, trivial and which admits finite coding hasnonisomorphic countable models.Combined with the corresponding result or superstable theories from [4] our result confirms the Vaught conjecture for trivial theories which admit finite coding.

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Generalized prime models

Journal of Symbolic Logic ◽

10.2307/2272463 ◽

1971 ◽

Vol 36 (4) ◽

pp. 593-606 ◽

Cited By ~ 1

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.

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Nonarithmetical ℵ0-categorical theories with recursive models

Journal of Symbolic Logic ◽

10.2307/2275253 ◽

1994 ◽

Vol 59 (1) ◽

pp. 106-112 ◽

Cited By ~ 9

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.

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THORN-FORKING AS LOCAL FORKING

Journal of Mathematical Logic ◽

10.1142/s0219061309000823 ◽

2009 ◽

Vol 09 (01) ◽

pp. 21-38 ◽

Cited By ~ 5

Author(s):

HANS ADLER

Keyword(s):

Order Theory ◽

Theoretical Interpretation ◽

First Order ◽

Closed Sets ◽

Local Character ◽

First Order Theory ◽

Simple Lattice ◽

Complete Theories

We introduce the notion of a preindependence relation between subsets of the big model of a complete first-order theory, an abstraction of the properties which numerous concrete notions such as forking, dividing, thorn-forking, thorn-dividing, splitting or finite satisfiability share in all complete theories. We examine the relation between four additional axioms (extension, local character, full existence and symmetry) that one expects of a good notion of independence. We show that thorn-forking can be described in terms of local forking if we localize the number k in Kim's notion of "dividing with respect to k" (using Ben-Yaacov's "k-inconsistency witnesses") rather than the forking formulas. It follows that every theory with an M-symmetric lattice of algebraically closed sets (in Teq) is rosy, with a simple lattice theoretical interpretation of thorn-forking.

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Models with second order properties in successors of singulars

Journal of Symbolic Logic ◽

10.2307/2275020 ◽

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.

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

Fundamenta Informaticae ◽

10.3233/fi-1982-53-406 ◽

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.

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Prime models and almost decidability

Journal of Symbolic Logic ◽

10.1017/s0022481200031273 ◽

1986 ◽

Vol 51 (2) ◽

pp. 412-420 ◽

Cited By ~ 2

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)].

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A complicated ω-stable depth 2 theory

Journal of Symbolic Logic ◽

10.2178/jsl/1294170989 ◽

2011 ◽

Vol 76 (1) ◽

pp. 47-65 ◽

Cited By ~ 2

Author(s):

Martin Koerwien

Keyword(s):

Order Theory ◽

The Other ◽

First Order ◽

First Order Theory ◽

Other Hand ◽

Countable Models

AbstractWe present a countable complete first order theory T which is model theoretically very well behaved: it eliminates quantifiers, is ω-stable, it has NDOP and is shallow of depth two. On the other hand, there is no countable bound on the Scott heights of its countable models, which implies that the isomorphism relation for countable models is not Borel.

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