A note on finitely generated models

1983 ◽  
Vol 48 (1) ◽  
pp. 163-166
Author(s):  
Anand Pillay
Keyword(s):  
Countable Model ◽  
Stable Theory ◽  
Prime Model ◽  
Finitely Generated ◽  
First Order ◽  
Elementary Chain ◽  
Skolem Functions ◽  
Order Language ◽  
Finite Set

A model M (of a countable first order language) is said to be finitely generated if it is prime over a finite set, namely if there is a finite tuple ā in M such that (M, ā) is a prime model of its own theory. Similarly, if A ⊂ M, then M is said to be finitely generated overA if there is finite ā in M such that M is prime over A ⋃ ā. (Note that if Th(M) has Skolem functions, then M being prime over A is equivalent to M being generated by A in the usual sense, that is, M is the closure of A under functions of the language.) We show here that if N is ā model of an ω-stable theory, M ≺ N, M is finitely generated, and N is finitely generated over M, then N is finitely generated. A corollary is that any countable model of an ω-stable theory is the union of an elementary chain of finitely generated models. Note again that all this is trivial if the theory has Skolem functions.The result here strengthens the results in [3], where we show the same thing but assuming in addition that the theory is either nonmultidimensional or with finite αT. However the proof in [3] for the case αT finite actually shows the following which does not assume ω-stability): Let A be atomic over a finite set, tp(ā / A) have finite Cantor-Bendixson rank, and B be atomic over A ⋃ ā. Then B is atomic over a finite set.

scholarly journals COUNTABLE MODELS OF THE THEORIES OF BALDWIN–SHI HYPERGRAPHS AND THEIR REGULAR TYPES

2019 ◽  
Vol 84 (3) ◽  
pp. 1007-1019
Author(s):  
DANUL K. GUNATILLEKA
Keyword(s):  
Large Body ◽  
Countable Model ◽  
Stable Theory ◽  
Generic Structure ◽  
Elementary Chain ◽  
Regular Type ◽  
Image Position ◽  
Original Class ◽  
Countable Models

AbstractWe continue the study of the theories of Baldwin–Shi hypergraphs from [5]. Restricting our attention to when the rank δ is rational valued, we show that each countable model of the theory of a given Baldwin–Shi hypergraph is isomorphic to a generic structure built from some suitable subclass of the original class used in the construction. We introduce a notion of dimension for a model and show that there is a an elementary chain $\left\{ {\mathfrak{M}_\beta :\beta \leqslant \omega } \right\}$ of countable models of the theory of a fixed Baldwin–Shi hypergraph with $\mathfrak{M}_\beta \preccurlyeq \mathfrak{M}_\gamma $ if and only if the dimension of $\mathfrak{M}_\beta $ is at most the dimension of $\mathfrak{M}_\gamma $ and that each countable model is isomorphic to some $\mathfrak{M}_\beta $. We also study the regular types that appear in these theories and show that the dimension of a model is determined by a particular regular type. Further, drawing on a large body of work, we use these structures to give an example of a pseudofinite, ω-stable theory with a nonlocally modular regular type, answering a question of Pillay in [11].

Describing Groups

2007 ◽  
Vol 13 (3) ◽  
pp. 305-339 ◽  
Author(s):  
André Nies
Keyword(s):  
Finite Automaton ◽  
Finite Automata ◽  
Finite Alphabet ◽  
Prime Model ◽  
Finitely Generated ◽  
Ring Of Integers ◽  
Group Operation ◽  
First Order

AbstractTwo ways of describing a group are considered. 1. A group is finite-automaton presentable if its elements can be represented by strings over a finite alphabet, in such a way that the set of representing strings and the group operation can be recognized by finite automata. 2. An infinite f.g. group is quasi-finitely axiomatizable if there is a description consisting of a single first-order sentence, together with the information that the group is finitely generated. In the first part of the paper we survey examples of FA-presentable groups, but also discuss theorems restricting this class. In the second part, we give examples of quasi-finitely axiomatizable groups, consider the algebraic content of the notion, and compare it to the notion of a group which is a prime model. We also show that if a structure is bi-interpretable in parameters with the ring of integers, then it is prime and quasi-finitely axiomatizable.

Decidable properties of finite sets of equations in trivial languages

1984 ◽  
Vol 49 (4) ◽  
pp. 1333-1338
Author(s):  
Cornelia Kalfa
Keyword(s):  
Equational Theory ◽  
Order Theory ◽  
Operation Symbol ◽  
First Order ◽  
First Order Theory ◽  
Order Language ◽  
Finite Set ◽  
Joint Embedding ◽  
Algebraic Language ◽  
Joint Embedding Property

In [4] I proved that in any nontrivial algebraic language there are no algorithms which enable us to decide whether a given finite set of equations Σ has each of the following properties except P2 (for which the problem is open):P0(Σ) = the equational theory of Σ is equationally complete.P1(Σ) = the first-order theory of Σ is complete.P2(Σ) = the first-order theory of Σ is model-complete.P3(Σ) = the first-order theory of the infinite models of Σ is complete.P4(Σ) = the first-order theory of the infinite models of Σ is model-complete.P5(Σ) = Σ has the joint embedding property.In this paper I prove that, in any finite trivial algebraic language, such algorithms exist for all the above Pi's. I make use of Ehrenfeucht's result [2]: The first-order theory generated by the logical axioms of any trivial algebraic language is decidable. The results proved here are part of my Ph.D. thesis [3]. I thank Wilfrid Hodges, who supervised it.Throughout the paper is a finite trivial algebraic language, i.e. a first-order language with equality, with one operation symbol f of rank 1 and at most finitely many constant symbols.

Theories without countable models

1972 ◽  
Vol 37 (3) ◽  
pp. 562-568
Author(s):  
Andreas Blass
Keyword(s):  
Standard Model ◽  
Countable Model ◽  
Compactness Theorem ◽  
Complete Theory ◽  
Natural Numbers ◽  
Axiom Schema ◽  
First Order ◽  
Order Language ◽  
The Standard Model ◽  
Skolem Theorem

Consider the Löwenheim-Skolem theorem in the form: If a theory in a countable first-order language has a model, then it has a countable model. As is well known, this theorem becomes false if one omits the hypothesis that the language be countable, for one then has the following trivial counterexample.Example 1. Let the language have uncountably many constants, and let the theory say that they are unequal.To motivate some of our future definitions and to introduce some notation, we present another, less trivial, counterexample.Example 2. Let L0 be the language whose n-place predicate (resp. function) symbols are all the n-place predicates (resp. functions) on the set ω of natural numbers. Let be the standard model for L0; we use the usual notation Th() for its complete theory. Add to L0 a new constant e, and add to Th() an axiom schema saying that e is infinite. By the compactness theorem, the resulting theory T has models. However, none of its models are countable. Although this fact is well known, we sketch a proof in order to refer to it later.By [5, p. 81], there is a family {Aα ∣ < α < c} of infinite subsets of ω, the intersection of any two of which is finite.

Generalized quantifiers and elementary extensions of countable models

1977 ◽  
Vol 42 (3) ◽  
pp. 341-348 ◽  
Author(s):  
Małgorzata Dubiel
Keyword(s):  
Countable Model ◽  
Generalized Quantifiers ◽  
Natural Question ◽  
Elementary Extension ◽  
First Order ◽  
Transitive Model ◽  
Order Language ◽  
Image Position ◽  
Countable Models

Let L be a countable first-order language and L(Q) be obtained by adjoining an additional quantifier Q. Q is a generalization of the quantifier “there exists uncountably many x such that…” which was introduced by Mostowski in [4]. The logic of this latter quantifier was formalized by Keisler in [2]. Krivine and McAloon [3] considered quantifiers satisfying some but not all of Keisler's axioms. They called a formula φ(x) countable-like iffor every ψ. In Keisler's logic, φ(x) being countable-like is the same as ℳ⊨┐Qxφ(x). The main theorem of [3] states that any countable model ℳ of L[Q] has an elementary extension N, which preserves countable-like formulas but no others, such that the only sets definable in both N and M are those defined by formulas countable-like in M. Suppose C(x) in M is linearly ordered and noncountable-like but with countable-like proper segments. Then in N, C will have new elements greater than all “old” elements but no least new element — otherwise it will be definable in both models. The natural question is whether it is possible to use generalized quantifiers to extend models elementarily in such a way that a noncountable-like formula C will have a minimal new element. There are models and formulas for which it is not possible. For example let M be obtained from a minimal transitive model of ZFC by letting Qxφ(x) mean “there are arbitrarily large ordinals satisfying φ”.

An interpretation of “finite” modal first-order languages in classical second-order languages

1976 ◽  
Vol 41 (2) ◽  
pp. 337-340
Author(s):  
Scott K. Lehmann
Keyword(s):  
Possible Worlds ◽  
Second Order ◽  
Universal Quantifier ◽  
Possible World ◽  
Simple Interpretation ◽  
First Order ◽  
Order Language ◽  
Finite Set ◽  
Modal Semantics ◽  
Individual Variables

This note describes a simple interpretation * of modal first-order languages K with but finitely many predicates in derived classical second-order languages L(K) such that if Γ is a set of K-formulae, Γ is satisfiable (according to Kripke's 55 semantics) iff Γ* is satisfiable (according to standard (or nonstandard) second-order semantics).The motivation for the interpretation is roughly as follows. Consider the “true” modal semantics, in which the relative possibility relation is universal. Here the necessity operator can be considered a universal quantifier over possible worlds. A possible world itself can be identified with an assignment of extensions to the predicates and of a range to the quantifiers; if the quantifiers are first relativized to an existence predicate, a possible world becomes simply an assignment of extensions to the predicates. Thus the necessity operator can be taken to be a universal quantifier over a class of assignments of extensions to the predicates. So if these predicates are regarded as naming functions from extensions to extensions, the necessity operator can be taken as a string of universal quantifiers over extensions.The alphabet of a “finite” modal first-order language K shall consist of a non-empty countable set Var of individual variables, a nonempty finite set Pred of predicates, the logical symbols ‘¬’ ‘∧’, and ‘∧’, and the operator ‘◊’. The formation rules of K generate the usual Polish notations as K-formulae. ‘ν’, ‘ν1’, … range over Var, ‘P’ over Pred, ‘A’ over K-formulae, and ‘Γ’ over sets of K-formulae.

On spectra, and the negative solution of the decision problem for identities having a finite nontrivial model

1975 ◽  
Vol 40 (2) ◽  
pp. 186-196 ◽  
Author(s):  
Ralph Mckenzie
Keyword(s):  
Model Problem ◽  
Recursive Algorithm ◽  
Finite Model ◽  
Positive Direction ◽  
First Order ◽  
Sharp Form ◽  
Universal Sentence ◽  
Order Language ◽  
Finite Set ◽  
The Universe

An algorithm has been described by S. Burris [3] which decides if a finite set of identities, whose function symbols are of rank at most 1, has a finite, nontrivial model. (By “nontrivial” it is meant that the universe of the model has at least two elements.) As a consequence of some results announced in the abstracts [2] and [8], it is clear that if the restriction on the ranks of function symbols is relaxed somewhat, then this finite model problem is no longer solvable by an algorithm, or at least not by a “recursive algorithm” as the term is used today.In this paper we prove a sharp form of this negative result; showing, by the way, that Burris' result is in a sense the best possible result in the positive direction. Our main result is that in a first order language whose only function or relation symbol is a 2-place function symbol (the language of groupoids), the set of identities that have no nontrivial model, is recursively inseparable from the set of identities such that the sentence has a finite model. As a corollary, we have that each of the following problems, restricted to sentences defined in the language of groupoids, is algorithmically unsolvable: (1) to decide if an identity has a finite nontrivial model; (2) to decide if an identity has a nontrivial model; (3) to decide if a universal sentence has a finite model; (4) to decide if a universal sentence has a model. We note that the undecidability of (2) was proved earlier by McNulty [13, Theorem 3.6(i)], improving results obtained by Murskiǐ [14] and by Perkins [17]. The other parts of the corollary seem to be new.

Skolem functions and elementary embeddings

1977 ◽  
Vol 42 (1) ◽  
pp. 94-98 ◽  
Author(s):  
Julia F. Knight
Keyword(s):  
Elementary Embedding ◽  
Elementary Substructure ◽  
First Order ◽  
Skolem Function ◽  
Free Variables ◽  
Skolem Functions ◽  
Order Language ◽  
Special Cases ◽  
Elementary Embeddings ◽  
Positive Results

Let L be an elementary first order language. Let be an L-structure, and let φ be an L-formula with free variables u1, …, un, and υ. A Skolem function for φ on is an n-ary operation f on such that for all . If is an elementary substructure of , then an n-ary operation f on is said to preserve the elementary embedding of into if f(x)∈ for all x ∈ n, and (, f ∣n) ≺ (, f). Keisler asked the following question:Problem 1. If and are L-structures such that ≺ , and if φ (u, υ) is an L-formula (with appropriate free variables), must there be a Skolem function for φ on which preserves the elementary embedding?Payne [6] gave a counterexample in which the language L is uncountable. In [3], [5], the author announced the existence of an example in which L is countable but the structures and are uncountable. The construction of the example will be given in this paper. Keisler's problem is still open in case both the language and the structures are required to be countable. Positive results for some special cases are given in [4].The following variant of Keisler's question was brought to the author's attention by Peter Winkler:Problem 2. If L is a countable language, a countable L-structure, and φ(u, υ) an L-formula, must there be a Skolem function f for φ on such that for every countable elementary extension of , there is an extension of f which preserves the elementary embedding of into ?

Truth Definitions, Skolem Functions and Axiomatic Set Theory

1998 ◽  
Vol 4 (3) ◽  
pp. 303-337 ◽  
Author(s):  
Jaakko Hintikka
Keyword(s):  
Set Theory ◽  
Model Theory ◽  
Order Logic ◽  
First Order Logic ◽  
First Order ◽  
Negative Results ◽  
Skolem Functions ◽  
Order Language ◽  
Definition Of ◽  
Axiomatic Set Theory

§1. The mission of axiomatic set theory. What is set theory needed for in the foundations of mathematics? Why cannot we transact whatever foundational business we have to transact in terms of our ordinary logic without resorting to set theory? There are many possible answers, but most of them are likely to be variations of the same theme. The core area of ordinary logic is by a fairly common consent the received first-order logic. Why cannot it take care of itself? What is it that it cannot do? A large part of every answer is probably that first-order logic cannot handle its own model theory and other metatheory. For instance, a first-order language does not allow the codification of the most important semantical concept, viz. the notion of truth, for that language in that language itself, as shown already in Tarski (1935). In view of such negative results it is generally thought that one of the most important missions of set theory is to provide the wherewithal for a model theory of logic. For instance Gregory H. Moore (1994, p. 635) asserts in his encyclopedia article “Logic and set theory” thatSet theory influenced logic, both through its semantics, by expanding the possible models of various theories and by the formal definition of a model; and through its syntax, by allowing for logical languages in which formulas can be infinite in length or in which the number of symbols is uncountable.

Model-theoretic properties characterizing Peano arithmetic

1991 ◽  
Vol 56 (3) ◽  
pp. 949-963 ◽  
Author(s):  
Richard Kaye
Keyword(s):  
Peano Arithmetic ◽  
Countable Model ◽  
Elementary Extension ◽  
First Order ◽  
Complete Extension ◽  
Order Language

AbstractLet = {0,1, +,·,<} be the usual first-order language of arithmetic. We show that Peano arithmetic is the least first-order -theory containing IΔ0 + exp such that every complete extension T of it has a countable model K satisfying(i) K has no proper elementary substructures, and(ii) whenever L ≻ K is a countable elementary extension there is and such that .Other model-theoretic conditions similar to (i) and (ii) are also discussed and shown to characterize Peano arithmetic.

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