Nonarithmetical ℵ0-categorical theories with recursive models

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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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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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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Models of countable theories

A First Course in Logic ◽

10.1093/oso/9780198529804.003.0010 ◽

2004 ◽

Author(s):

Shawn Hedman

Keyword(s):

Order Theory ◽

Cardinal Function ◽

First Order ◽

First Order Theory ◽

Free Variables ◽

Basic Facts ◽

Small Models

We define and study types of a complete first-order theory T. This concept allows us to refine our analysis of Mod(T). If T has few types, then Mod(T) contains a uniquely defined smallest model that can be elementarily embedded into any structure of Mod(T). We investigate the various properties of these small models in Section 6.3. In Section 6.4, we consider the “big” models of Mod(T). For any theory, the number of types is related to the number of models of the theory. For any cardinal κ, I(T, κ) denotes the number of models in Mod(T) of size κ. We prove two basic facts regarding this cardinal function. In Section 6.5, we show that if T has many types, then I(T, κ) takes on its maximal possible value of 2κ for each infinite κ. In Section 6.6, we prove Vaught’s theorem stating that I(T, ℵ0) cannot equal 2. All formulas are first-order formulas. All theories are sets of first-order sentences. For any structure M, we conveniently refer to an n-tuple of elements from the underlying set of M as an “n-tuple of M.” The notion of a type extends the notion of a theory to include formulas and not just sentences. Whereas theories describe structures, types describe elements within a structure. Definition 6.1 Let M be a ν-structure and let ā = (a1, . . . , an) be an n-tuple of M. The type of ā in M, denoted tpM(ā), is the set of all ν-formulas φ having free variables among x1, . . . , xn that hold in M when each xi in is replaced by ai. More concisely, but less precisely: If ā is an n-tuple, then each formula in tpM(ā) contains at most n free variables but may contain fewer. In particular, the type of an n-tuple contains sentences. For any structure M and tuple ā of M, tpM(ā) contains Th(M) as a subset. The set tpM(ā) provides the complete first-order description of the tuple ā and how it sits in M.

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