A Note on the Realization of Types

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.

Theories with finitely many models

10.1017/s0022481200031236 ◽

1986 ◽

Vol 51 (2) ◽

pp. 374-376 ◽

Cited By ~ 1

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.

Almost strongly minimal theories. I

10.2307/2272733 ◽

1972 ◽

Vol 37 (3) ◽

pp. 487-493 ◽

Cited By ~ 5

Author(s):

John T. Baldwin

Keyword(s):

Algebraic Closure ◽

Natural Extension ◽

Stone Space ◽

First Order ◽

Order Language ◽

Simple Class ◽

The Universe ◽

Natural Expansion ◽

Minimal Formula

In [1] the notions of strongly minimal formula and algebraic closure were applied to the study of ℵ1-categorical theories. In this paper we study a particularly simple class of ℵ1-categorical theories. We characterize this class in terms of the analysis of the Stone space of models of T given by Morley [3].We assume familiarity with [1] and [3], but for convenience we list the principal results and definitions from those papers which are used here. Our notation is the same as in [1] with the following exceptions.We deal with a countable first order language L. We may extend the language L in several ways. If is an L-structure, there is a natural extension of L obtained by adjoining to L a constant a for each (the universe of ). For each sentence A(a1, …, an) ∈ L(A) we say satisfies A(a1, …, an) and write if in Shoenfield's notation If is an L-structure and X is a subset of , then L(X) is the language obtained by adjoining to L a name x for each is the natural expansion of to an L(X)-structure. A structure is an inessential expansion [4, p. 141] of an L-structure if for some .

An axiomatic system for the first order language with an equi-cardinality quantifier

10.2307/2269698 ◽

1966 ◽

Vol 31 (4) ◽

pp. 633-640 ◽

Cited By ~ 3

Author(s):

Mitsuru Yasuhara

Keyword(s):

Rational Numbers ◽

Axiomatic System ◽

Universal Quantifier ◽

Real Numbers ◽

Relational System ◽

First Order ◽

Order Language ◽

Relational Systems ◽

Finite Domains ◽

The equi-cardinality quantifier1 to be used in this article, written as Qx, is characterised by the following semantical rule: A formula QxA(x) is true in a relational system exactly when the cardinality of the set consisting of these elements which make A(x) true is the same as that of the universe. For instance, QxN(x) is true in 〈Rt, N〉 but false in 〈Rl, N〉 where Rt, Rl, and N are the sets of rational numbers, real numbers, and natural numbers, respectively. We notice that in finite domains the equi-cardinality quantifier is the same as the universal quantifier. For this reason, all relational systems considered in the following are assumed infinite.

Theories without countable models

10.2307/2272744 ◽

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.

Noninitial segments of the α-degrees

10.2307/2273029 ◽

1973 ◽

Vol 38 (3) ◽

pp. 368-388 ◽

Cited By ~ 2

Author(s):

John M. Macintyre

Keyword(s):

Distributive Lattice ◽

Initial Segment ◽

Elementary Theory ◽

Recursion Theory ◽

Partial Ordering ◽

First Order ◽

Order Language ◽

Suitable Notion ◽

The Universe ◽

Degrees Of Unsolvability

Let α be an admissible ordinal and let L be the first order language with equality and a single binary relation ≤. The elementary theory of the α-degrees is the set of all sentences of L which are true in the universe of the α-degrees when ≤ is interpreted as the partial ordering of the α-degrees. Lachlan [6] showed that the elementary theory of the ω-degrees is nonaxiomatizable by proving that any countable distributive lattice with greatest and least members can be imbedded as an initial segment of the degrees of unsolvability. This paper deals with the extension of these results to α-recursion theory for an arbitrary countable admissible α > ω. Given α, we construct a set A with α-degree a such that every countable distributive lattice with greatest and least member is order isomorphic to a segment of α-degrees {d ∣ a ≤αd≤αb} for some α-degree b. As in [6] this implies that the elementary theory of the α-degrees is nonaxiomatizable and hence undecidable.A is constructed in §2. A is a set of integers which is generic with respect to a suitable notion of forcing. Additional applications of such sets are summarized at the end of the section. In §3 we define the notion of a tree and construct a particular tree T0 which is weakly α-recursive in A. Using T0 we can apply the techniques of [6] and [2] to α-recursion theory. In §4 we reduce our main results to three technical lemmas concerning systems of trees. These lemmas are proved in §5.

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

10.2307/2271899 ◽

1975 ◽

Vol 40 (2) ◽

pp. 186-196 ◽

Cited By ~ 41

Author(s):

Ralph Mckenzie

Keyword(s):

Model Problem ◽

Recursive Algorithm ◽

Finite Model ◽

Positive Direction ◽

First Order ◽

Sharp Form ◽

Universal Sentence ◽

Order Language ◽

Finite Set ◽

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.

Logic of reduced power structures

10.2307/2273319 ◽

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].

A form of feasible interpolation for constant depth Frege systems

10.2178/jsl/1268917504 ◽

2010 ◽

Vol 75 (2) ◽

pp. 774-784 ◽

Cited By ~ 4

Author(s):

Jan Krajíček

Keyword(s):

Constant Depth ◽

First Order ◽

Order Language ◽

Frege Systems ◽

Image Position ◽

The Universe ◽

Dnf Formula

AbstractLet L be a first-order language and Φ and Ψ two L-sentences that cannot be satisfied simultaneously in any finite L-structure. Then obviously the following principle ChainL,Φ,Ψ(n, m) holds: For any chain of finite L-structures C1, …, Cm with the universe [n] one of the following conditions must fail:For each fixed L and parameters n, m the principle ChainL,Φ,Ψ(n,m) can be encoded into a propositional DNF formula of size polynomial in n, m.For any language L containing only constants and unary predicates we show that there is a constant CL such that the following holds: If a constant depth Frege system in DeMorgan language proves ChainL,Φ,Ψ(n, cL . n) by a size s proof then the class of finite L-structures with universe [n] satisfying Φ can be separated from the class of those L-structures on [n] satisfying ψ by a depth 3 formula of size 2log(S)O(1) and with bottom fan-in log(S)O(1).

Stratification and cut-elimination

10.2307/2274915 ◽

1991 ◽

Vol 56 (1) ◽

pp. 213-226 ◽

Cited By ~ 5

Author(s):

Marcel Crabbé

Keyword(s):

Set Theory ◽

Type Theory ◽

Cut Elimination ◽

Simple Type ◽

First Order ◽

Similar Reason ◽

Separation Axioms ◽

Simple Type Theory ◽

Order Language ◽

In this paper, we show the normalization of proofs of NF (Quine's New Foundations; see [15]) minus extensionality. This system, called SF (Stratified Foundations) differs in many respects from the associated system of simple type theory. It is written in a first order language and not in a multi-sorted one, and the formulas need not be stratifiable, except in the instances of the comprehension scheme. There is a universal set, but, for a similar reason as in type theory, the paradoxical sets cannot be formed.It is not immediately apparent, however, that SF is essentially richer than type theory. But it follows from Specker's celebrated result (see [16] and [4]) that the stratifiable formula (extensionality → the universe is not well-orderable) is a theorem of SF.It is known (see [11]) that this set theory is consistent, though the consistency of NF is still an open problem.The connections between consistency and cut-elimination are rather loose. Cut-elimination generally implies consistency. But the converse is not true. In the case of set theory, for example, ZF-like systems, though consistent, cannot be freed of cuts because the separation axioms allow the formation of sets from unstratifiable formulas. There are nevertheless interesting partial results obtained when restrictions are imposed on the removable cuts (see [1] and [9]). The systems with stratifiable comprehension are the only known set-theoretic systems that enjoy full cut-elimination.

A remark on infinitary languages

10.2307/2269955 ◽

1971 ◽

Vol 36 (3) ◽

pp. 461-462 ◽

Cited By ~ 1

Author(s):

Jörg Flum

Keyword(s):

First Order ◽

Order Language ◽

In his paper [1] Chang provides among other things answers to questions of the following type: Given two models and of powers α and β, respectively, what is the least λ such that implies His proofs are by induction on the quantifier rank of formulas and they use an idea which in the case of ordinary first-order language goes back to Ehrenfeucht and Fraïssé. But, as we show, one can easily prove that if λ is big compared with κ and with the cardinality of the universe of the structure , then every L∞κ-formula is equivalent modulo the set of all Lλκ-sentences which hold in to a Lλκ-formula. From this, Chang's results follow immediately. The same method can be applied to similar problems concerning generalized languages.