Theories with finitely many models

1986 ◽  
Vol 51 (2) ◽  
pp. 374-376 ◽  
Author(s):  
Simon Thomas
Keyword(s):  
Natural Number ◽  
Complete Theory ◽  
Elementary Equivalence ◽  
First Order ◽  
Finite Models ◽  
Simple Observation ◽  
Finite Structure ◽  
Order Language ◽  
Complete Theories

If L is a first order language and n is a natural number, then Ln is the set of formulas which only make use of the variables x1,…,xn. While every finite structure is determined up to isomorphism by its theory in L, the same is no longer true in Ln. This simple observation is the source of a number of intriguing questions. For example, Poizat [2] has asked whether a complete theory in Ln which has at least two nonisomorphic finite models must necessarily also have an infinite one. The purpose of this paper is to present some counterexamples to this conjecture.Theorem. For each n ≤ 3 there are complete theories in L2n−2andL2n−1having exactly n + 1 models.In our notation and definitions, we follow Poizat [2]. To test structures for elementary equivalence in Ln, we shall use the modified Ehrenfeucht-Fraïssé games of Immerman [1]. For convenience, we repeat his definition here.Suppose that L is a purely relational language, each of the relations having arity at most n. Let and ℬ be two structures for L. Define the Ln game on and ℬ as follows. There are two players, I and II, and there are n pairs of counters a1, b1, …, an, bn. On each move, player I picks up any of the counters and places it on an element of the appropriate structure.

On the elementary equivalence of automorphism groups of Boolean algebras; downward Skolem Löwenheim theorems and compactness of related quantifiers

1980 ◽  
Vol 45 (2) ◽  
pp. 265-283 ◽  
Author(s):  
Matatyahu Rubin ◽  
Saharon Shelah
Keyword(s):  
Boolean Algebra ◽  
Automorphism Groups ◽  
Order Logic ◽  
Boolean Algebras ◽  
First Order Logic ◽  
Elementary Equivalence ◽  
First Order ◽  
Finite Models ◽  
Countably Compact

AbstractTheorem 1. (◊ℵ1,) If B is an infinite Boolean algebra (BA), then there is B1, such that ∣ Aut (B1) ≤∣B1∣ = ℵ1 and 〈B1, Aut (B1)〉 ≡ 〈B, Aut(B)〉.Theorem 2. (◊ℵ1) There is a countably compact logic stronger than first-order logic even on finite models.This partially answers a question of H. Friedman. These theorems appear in §§1 and 2.Theorem 3. (a) (◊ℵ1) If B is an atomic ℵ-saturated infinite BA, Ψ Є Lω1ω and 〈B, Aut (B)〉 ⊨Ψ then there is B1, Such that ∣Aut(B1)∣ ≤ ∣B1∣ =ℵ1, and 〈B1, Aut(B1)〉⊨Ψ. In particular if B is 1-homogeneous so is B1. (b) (a) holds for B = P(ω) even if we assume only CH.

On first-order sentences without finite models

2004 ◽  
Vol 69 (2) ◽  
pp. 329-339 ◽  
Author(s):  
Marko Djordjević
Keyword(s):  
Binary Relation ◽  
Function Symbol ◽  
Complete Theory ◽  
Order Property ◽  
Unary Function ◽  
First Order ◽  
Finite Models

We will mainly be concerned with a result which refutes a stronger variant of a conjecture of Macpherson about finitely axiomatizable ω-categorical theories. Then we prove a result which implies that the ω-categorical stable pseudoplanes of Hrushovski do not have the finite submodel property.Let's call a consistent first-order sentence without finite models an axiom of infinity. Can we somehow describe the axioms of infinity? Two standard examples are:ϕ1: A first-order sentence which expresses that a binary relation < on a nonempty universe is transitive and irreflexive and that for every x there is y such that x < y.ϕ2: A first-order sentence which expresses that there is a unique x such that, (0) for every y, s(y) ≠ x (where s is a unary function symbol),and, for every x, if x does not satisfy (0) then there is a unique y such that s(y) = x.Every complete theory T such that ϕ1 ϵ T has the strict order property (as defined in [10]), since the formula x < y will have the strict order property for T. Let's say that if Ψ is an axiom of infinity and every complete theory T with Ψ ϵ T has the strict order property, then Ψ has the strict order property.Every complete theory T such that ϕ2 ϵ T is not ω-categorical. This is the case because a complete theory T without finite models is ω-categorical if and only if, for every 0 < n < ω, there are only finitely many formulas in the variables x1,…,xn, up to equivalence, in any model of T.

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.

On the ring of differentially-algebraic entire functions

1992 ◽  
Vol 57 (2) ◽  
pp. 449-451 ◽  
Author(s):  
Lee A. Rubel
Keyword(s):  
Differential Equation ◽  
Finite Length ◽  
Complex Variable ◽  
Entire Functions ◽  
Algebraic Differential Equation ◽  
Elementary Equivalence ◽  
First Order ◽  
Order Language ◽  
Image Position

Let be the ring of all entire functions of one complex variable, and let DA be the subring of those entire functions that are differentially algebraic (DA); that is, they satisfy a nontrivial algebraic differential equation.where P is a non-identically-zero polynomial in its n + 2 variables. It seems not to be known whether DA is elementarily equivalent to . This would mean that DA and have exactly the same true statements about them, in the first-order language of rings. (Roughly speaking, a sentence about a ring R is first-order if it has finite length and quantifies only over elements (i.e., not subsets or functions or relations) of R.) It follows from [NAN] that DA and are not isomorphic as rings, but this does not answer the question of elementary equivalence.

scholarly journals Undecidable properties of finite sets of equations

1976 ◽  
Vol 41 (3) ◽  
pp. 589-604 ◽  
Author(s):  
George F. McNulty
Keyword(s):  
Decision Problems ◽  
Equational Logic ◽  
Relation Algebras ◽  
Finite Sets ◽  
Operation Symbol ◽  
First Order ◽  
Finite Models ◽  
Order Language

Though equations are among the simplest sentences available in a first order language, many of the most familiar notions from algebra can be expressed by sets of equations. It is the task of this paper to expose techniques and theorems that can be used to establish that many collections of finite sets of equations characterized by common algebraic or logical properties fail to be recursive. The following theorem is typical.Theorem. In a language provided with an operation symbol of rank at least two, the collection of finite irredundant sets of equations is not recursive.Theorems of this kind are part of a pattern of research into decision problems in equational logic. This pattern finds its origins in the works of Markov [8] and Post [20] and in Tarski's development of the theory of relation algebras; see Chin [1], Chin and Tarski [2], and Tarski [23]. The papers of Mal′cev [7] and Perkins [16] are more directly connected with the present paper, which includes generalization of much of Perkins' work as well as extensions of a theorem of D. Smith [22]. V. L. Murskii [14] contains some of the results below discovered independently. Not all known results concerning undecidable properties of finite sets of equations seem to be susceptible to the methods presented here. R. McKenzie, for example, shows in [9] that for a language with an operation symbol of rank at least two, the collection of finite sets of equations with nontrivial finite models is not recursive. D. Pigozzi has extended and elaborated the techniques of this paper in [17], [18], and [19] to obtain new results concerning undecidable properties, particularly those of algebraic character.

A Note on the Realization of Types

1980 ◽  
Vol 23 (1) ◽  
pp. 95-98
Author(s):  
Alan Adamson
Keyword(s):  
Complete Theory ◽  
First Order ◽  
Order Language ◽  
The Universe

Let L be a countable first-order language and T a fixed complete theory in L. If is a model of T, is an n-sequence of variables, and ā=〈a1,…, an〉 is an n-sequence of elements of M, the universe of , we let where ranges over formulas of L containing freely at most the variables υ1,…υn. ā is said to realize in We let be where is the sequence of the first n variables of L.

Logic of reduced power structures

1983 ◽  
Vol 48 (1) ◽  
pp. 53-59
Author(s):  
G.C. Nelson
Keyword(s):  
Boolean Algebra ◽  
Complete Theory ◽  
Power Structures ◽  
Index Set ◽  
First Order ◽  
Order Language ◽  
Power Set ◽  
Definition Of ◽  
Proper Filter ◽  
Horn Sentence

We start with the framework upon which this paper is based. The most useful reference for these notions is [2]. For any nonempty index set I and any proper filter D on S(I) (the power set of I), we denote by I/D the reduced power of modulo D as defined in [2, pp. 167–169]. The first-order language associated with I/D will always be the same language as associated with . We denote the 2-element Boolean algebra 〈{0, 1}, ⋂, ⋃, c, 0, 1〉 by 2 and 2I/D denotes the reduced power of it modulo D. We point out the intimate connection between the structures I/D and 2I/D given in [2, pp. 341–345]. Moreover, we assume as known the definition of Horn formula and Horn sentence as given in [2, p. 328] along with the fundamental theorem that φ is a reduced product sentence iff φ is provably equivalent to a Horn sentence [2, Theorem 6.2.5/ (iff φ is a 2-direct product sentence and a reduced power sentence [2, Proposition 6.2.6(ii)]). For a theory T(any set of sentences), ⊨ T denotes that is a model of T.In addition to the above we assume as known the elementary characteristics (due to Tarski) associated with a complete theory of a Boolean algebra, and we adopt the notation 〈n, p, q〉 of [3], [10], or [6] to denote such an elementary characteristic or the corresponding complete theory. We frequently will use Ershov's theorem which asserts that for each 〈n, p, q〉 there exist an index set I and filter D such that 2I/D ⊨ 〈n, p, q〉 [3] or [2, Lemma 6.3.21].

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.

Some remarks on definable equivalence relations in O-minimal structures

1986 ◽  
Vol 51 (3) ◽  
pp. 709-714 ◽  
Author(s):  
Anand Pillay
Keyword(s):  
Equivalence Relation ◽  
Linear Order ◽  
Finite Union ◽  
Equivalence Relations ◽  
Complete Theory ◽  
First Order ◽  
Definable Set ◽  
Order Language ◽  
Minimal Structure ◽  
Definable Sets

Let M be an O-minimal structure. We use our understanding, acquired in [KPS], of the structure of definable sets of n-tuples in M, to study definable (in M) equivalence relations on Mn. In particular, we show that if E is an A-definable equivalence relation on Mn (A ⊂ M) then E has only finitely many classes with nonempty interior in Mn, each such class being moreover also A-definable. As a consequence, we are able to give some conditions under which an O-minimal theory T eliminates imaginaries (in the sense of Poizat [P]).If L is a first order language and M an L-structure, then by a definable set in M, we mean something of the form X ⊂ Mn, n ≥ 1, where X = {(a1…,an) ∈ Mn: M ⊨ϕ(ā)} for some formula ∈ L(M). (Here L(M) means L together with names for the elements of M.) If the parameters from come from a subset A of M, we say that X is A-definable.M is said to be O-minimal if M = (M, <,…), where < is a dense linear order with no first or last element, and every definable set X ⊂ M is a finite union of points, and intervals (a, b) (where a, b ∈ M ∪ {± ∞}). (This notion is as in [PS] except here we demand the underlying order be dense.) The complete theory T is said to be O-minimal if every model of T is O-minimal. (Note that in [KPS] it is proved that if M is O-minimal, then T = Th(M) is O-minimal.) In the remainder of this section and in §2, M will denote a fixed but arbitrary O-minimal structure. A,B,C,… will denote subsets of M.

On the Probabilistic Nature of Observable Phenomena

2019 ◽  
Author(s):  
Muhammad Ali
Keyword(s):  
Quantum Mechanics ◽  
Complete Theory ◽  
Interpretation Of Quantum Mechanics ◽  
Multiple Realities ◽  
Lack Of Information ◽  
Complete Theories ◽  
Probabilistic Nature

This paper proposes a Gadenkan experiment named “Observer’s Dilemma”, to investigate the probabilistic nature of observable phenomena. It has been reasoned that probabilistic nature in, otherwise uniquely deterministic phenomena can be introduced due to lack of information of underlying governing laws. Through theoretical consequences of the experiment, concepts of ‘Absolute Complete’ and ‘Observably Complete” theories have been introduced. Furthermore, nature of reality being ‘absolute’ and ‘observable’ have been discussed along with the possibility of multiple realities being true for observer. In addition, certain aspects of quantum mechanics have been interpreted. It has been argued that quantum mechanics is an ‘observably complete’ theory and its nature is to give probabilistic predictions. Lastly, it has been argued that “Everettian - Many world” interpretation of quantum mechanics is very real and true in the framework of ‘observable nature of reality’, for humans.

Sign in / Sign up

Export Citation Format

Share Document