Weakly atomic-compact relational structures

1971 ◽  
Vol 36 (1) ◽  
pp. 129-140 ◽  
Author(s):  
G. Fuhrken ◽  
W. Taylor
Keyword(s):  
Abelian Group ◽  
Model Theory ◽  
Compact Abelian Group ◽  
Finite Subset ◽  
Relational Structure ◽  
Similarity Type ◽  
First Order ◽  
Relational Structures ◽  
Order Language ◽  
Weakly Atomic

A relational structure is called weakly atomic-compact if and only if every set Σ of atomic formulas (taken from the first-order language of the similarity type of augmented by a possibly uncountable set of additional variables as “unknowns”) is satisfiable in whenever every finite subset of Σ is so satisfiable. This notion (as well as some related ones which will be mentioned in §4) was introduced by J. Mycielski as a generalization to model theory of I. Kaplansky's notion of an algebraically compact Abelian group (cf. [5], [7], [1], [8]).

Martin's axiom in the model theory of LA

1980 ◽  
Vol 45 (1) ◽  
pp. 172-176
Author(s):  
W. Richard Stark
Keyword(s):  
Model Theory ◽  
Equivalence Theorem ◽  
Compactness Theorem ◽  
Compactness Result ◽  
First Order ◽  
Martin's Axiom ◽  
New Class ◽  
Martin’S Axiom ◽  
Order Language ◽  
Omitting Types

Working in ZFC + Martin's Axiom we develop a generalization of the Barwise Compactness Theorem which holds in languages of cardinality less than . Next, using this compactness theorem, an omitting types theorem for fewer than types is proved. Finally, in ZFC, we prove that this compactness result implies Martin's Axiom (the Equivalence Theorem). Our compactness theorem applies to a new class of theories—ccΣ-theories—which generalize the countable Σ-theories of Barwise's theorem. The Omitting Types Theorem and the Equivalence Theorem serve as examples illustrating the use of ccΣ-theories.Assume = (A, ε) or = (A, ε R1,…,Rm) where is admissible. L() is the first-order language with constants for elements of A and relation symbols for relations in . LA is A ⋂ L∞ω where the L of L∞ω is any language in A. A theory T in LA is consistent if there is no derivation in A of a contradiction from T. is LA with new constants ca for each a and A. The basic terms of consist of the constants of and the terms f(ca1,…,cam) built directly from constants using functions f of . The symbol t is used for basic terms. A theory T in LA is Σ if it is defined by a formula of L(). The formula φ⌝ is a logical equivalent of ¬φ defined by: (1) φ⌝ = ¬φ if φ is atomic; (2) (¬φ)⌝ = φ (3) (⋁φ∈Φ φ)⌝ = ⋀φ∈Φ φ⌝; (4) (⋀φ∈Φ φ) ⋁φ∈Φ φ⌝; (5) (∃χφ(x))⌝ ∀χφ⌝(x); ∀χφ(x))⌝ = ∃χφ⌝(x).

Probabilities on finite models

1976 ◽  
Vol 41 (1) ◽  
pp. 50-58 ◽  
Author(s):  
Ronald Fagin
Keyword(s):  
Rate Of Convergence ◽  
Finite Subset ◽  
Asymptotic Estimate ◽  
Predicate Symbol ◽  
Relational Structure ◽  
Finite Model ◽  
First Order ◽  
Finite Models ◽  
Model Following ◽  
Finite Set

Let be a finite set of (nonlogical) predicate symbols. By an -structure, we mean a relational structure appropriate for . Let be the set of all -structures with universe {1, …, n}. For each first-order -sentence σ (with equality), let μn(σ) be the fraction of members of for which σ is true. We show that μn(σ) always converges to 0 or 1 as n → ∞, and that the rate of convergence is geometrically fast. In fact, if T is a certain complete, consistent set of first-order -sentences introduced by H. Gaifman [6], then we show that, for each first-order -sentence σ, μn(σ) →n 1 iff T ⊩ ω. A surprising corollary is that each finite subset of T has a finite model. Following H. Scholz [8], we define the spectrum of a sentence σ to be the set of cardinalities of finite models of σ. Another corollary is that for each first-order -sentence a, either σ or ˜σ has a cofinite spectrum (in fact, either σ or ˜σ is “nearly always“ true).Let be a subset of which contains for each in exactly one structure isomorphic to . For each first-order -sentence σ, let νn(σ) be the fraction of members of which a is true. By making use of an asymptotic estimate [3] of the cardinality of and by our previously mentioned results, we show that vn(σ) converges as n → ∞, and that limn νn(σ) = limn μn(σ).If contains at least one predicate symbol which is not unary, then the rate of convergence is geometrically fast.

Categoricity regained

1976 ◽  
Vol 41 (3) ◽  
pp. 639-643 ◽  
Author(s):  
Erik Ellentuck
Keyword(s):  
Infinite Length ◽  
Modern Logic ◽  
Mathematical Structures ◽  
Similarity Type ◽  
First Order ◽  
Nonstandard Model ◽  
Partial Isomorphism ◽  
Order Language ◽  
Order Arithmetic ◽  
Universe Of Sets

One of the earliest goals of modern logic was to characterize familiar mathematical structures up to isomorphism by means of properties expressed in a first order language. This hope was dashed by Skolem's discovery (cf. [6]) of a nonstandard model of first order arithmetic. A theory T such that any two of its models are isomorphic is called categorical. It is well known that if T has any infinite models then T is not categorical. We shall regain categoricity by(i) enlarging our language so as to allow expressions of infinite length, and(ii) enlarging our class of isomorphisms so as to allow isomorphisms existing in some Boolean valued extension of the universe of sets.Let and be mathematical structures of the same similarity type where say R is binary on A. We write if f is an isomorphism of onto , and if there is an f such that . We say that P is a partial isomorphism of onto and write if P is a nonempty set of functions such that(i) if f ∈ P then dom(f) is a substructure of , rng(/f) is a substructure of , and f is an isomorphism of its domain onto its range, and(ii) if f ∈ P, a ∈ A, and b ∈ B then there exist g,h ∈ P, both extending f such that a ∈ dom(g) and b ∈ rng(h). Write if there is a P such that .

On countable fractions from an elementary class

1994 ◽  
Vol 59 (4) ◽  
pp. 1410-1413
Author(s):  
C. J. Ash
Keyword(s):  
Model Theory ◽  
Relation Symbol ◽  
First Order ◽  
Elementary Class ◽  
Free Variables ◽  
Order Language ◽  
Skolem Theorem ◽  
Elementary Result ◽  
Countable Models

The following fairly elementary result seems to raise possibilities for the study of countable models of a theory in a countable language. For the terminology of model theory we refer to [CK].Let L be a countable first-order language. Let L′ be the relational language having, for each formula φ of L and each sequence υ1,…,υn of variables including the free variables of φ, an n-ary relation symbol Pφ. For any L-structure and any formula Ψ(υ) of L, we define the Ψ-fraction of to be the L′-structure Ψ whose universe consists of those elements of satisfying Ψ(υ) and whose relations {Rφ}φϵL are defined by letting a1,…,an satisfy Rφ in Ψ if, and only if, a1,…, an satisfy φ in .An L-elementary class means the class of all L-structures satisfying each of some set of sentences of L. The countable part of an L-elementary class K means the class of all countable L-structures from K.Theorem. Let K be an L-elementary class and let Ψ(υ) be a formula of L. Then the class of countable Ψ-fractions of structures in K is the countable part of some L′-elementary class.Comment. By the downward Löwenheim-Skolem theorem, the countable Ψ-fractions of structures in K are the same as the Ψ-fractions of countable structures in K.Proof. We give a set Σ′ of L′-sentences whose countable models are exactly the countable Ψ-fractions of structures in K.

scholarly journals About Quotient Orders and Ordering Sequences

2017 ◽  
Vol 25 (2) ◽  
pp. 121-139
Author(s):  
Sebastian Koch
Keyword(s):  
Finite Subset ◽  
Finite Sequence ◽  
Relational Structure ◽  
Real Numbers ◽  
Natural Restriction ◽  
Relational Structures ◽  
The Third ◽  
Finite Sequences ◽  
Directed Walks ◽  
Ordered Pairs

Summary In preparation for the formalization in Mizar [4] of lotteries as given in [14], this article closes some gaps in the Mizar Mathematical Library (MML) regarding relational structures. The quotient order is introduced by the equivalence relation identifying two elements x, y of a preorder as equivalent if x ⩽ y and y ⩽ x. This concept is known (see e.g. chapter 5 of [19]) and was first introduced into the MML in [13] and that work is incorporated here. Furthermore given a set A, partition D of A and a finite-support function f : A → ℝ, a function Σf : D → ℝ, Σf (X)= ∑x∈X f(x) can be defined as some kind of natural “restriction” from f to D. The first main result of this article can then be formulated as: $$\sum\limits_{x \in A} {f(x)} = \sum\limits_{X \in D} {\Sigma _f (X)\left( { = \sum\limits_{X \in D} {\sum\limits_{x \in X} {f(x)} } } \right)} $$ After that (weakly) ascending/descending finite sequences (based on [3]) are introduced, in analogous notation to their infinite counterparts introduced in [18] and [13]. The second main result is that any finite subset of any transitive connected relational structure can be sorted as a ascending or descending finite sequence, thus generalizing the results from [16], where finite sequence of real numbers were sorted. The third main result of the article is that any weakly ascending/weakly descending finite sequence on elements of a preorder induces a weakly ascending/weakly descending finite sequence on the projection of these elements into the quotient order. Furthermore, weakly ascending finite sequences can be interpreted as directed walks in a directed graph, when the set of edges is described by ordered pairs of vertices, which is quite common (see e.g. [10]). Additionally, some auxiliary theorems are provided, e.g. two schemes to find the smallest or the largest element in a finite subset of a connected transitive relational structure with a given property and a lemma I found rather useful: Given two finite one-to-one sequences s, t on a set X, such that rng t ⊆ rng s, and a function f : X → ℝ such that f is zero for every x ∈ rng s \ rng t, we have ∑ f o s = ∑ f o t.

Some elementary degree-theoretic reasons why structures need similarity types

1986 ◽  
Vol 51 (3) ◽  
pp. 732-747
Author(s):  
T. G. McLaughlin
Keyword(s):  
Natural Numbers ◽  
Similarity Type ◽  
First Order ◽  
Order Language ◽  
Disjoint Sets ◽  
And Function ◽  
The Way

By a “partly numerical structure” (p.n.s.) we shall here mean a quadruple , where M is a set, ω = the natural numbers, ω ⊆ M, and are disjoint sets, is a set of relations (of various positive integral arities) on M, and is a set of functions (of various positive integral arities) with arguments and values in M. Thus, in calculated disharmony with common practice, we do not (except as noted below, in connection with naming the elements of ω) fix a similarity type as part of our notion of a “structure”. Suppose a finitary first-order language (with identity) has been specified, with constant symbols , n ∈ ω, and with exactly enough relation and function symbols of each arity to enable us to interpret in . We wish to consider the variation in the degree (relative to a fixed Gödel-numbering of ) of the complete -theory of as we vary the way in which elements of ∪ are assigned as interpretations to the relation and function symbols of . We shall in fact, therefore, be concerned exclusively with p.n.s.'s for which ∪ is countable. More: we assume to be such that we can effectively tell, uniformly in n > 0, exactly how many n-ary relations has and exactly how many n-ary functions has.

Some Obstacles to Duality in Topological Algebra

1982 ◽  
Vol 34 (1) ◽  
pp. 80-90 ◽  
Author(s):  
Paul Bankston
Keyword(s):  
Full Subcategory ◽  
Topological Algebra ◽  
Usual Sense ◽  
Similarity Type ◽  
First Order ◽  
Equivalence Of Categories ◽  
Order Language

0. Introduction. Functors form an equivalence of categories (see [8,]) if Γ(Φ(A)) ≅ A and Φ (Γ(B)) ≅ B naturally for all objects A from and B from . Letting denote the opposite of we say that and are dual if there is an equivalence between and .Let τ be a similarity type of finitary operation symbols. We let Lτ denote the first order language (with equality) using nonlogical symbols from τ, and consider the class of all algebras of type τ as a category by declaring the morphisms to be all homomorphisms in the usual sense (i.e., those functions preserving the atomic sentences of Lτ). If is a class in (i.e., and is closed under isomorphism), we view as a full subcategory of , and we define the order of to be the number of symbols occurring in τ.

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.

Semantics for relevant logics

1972 ◽  
Vol 37 (1) ◽  
pp. 159-169 ◽  
Author(s):  
Alasdair Urquhart
Keyword(s):  
Modal Logic ◽  
Relational Structure ◽  
Material Implication ◽  
First Order ◽  
Relevant Logics ◽  
Order Language ◽  
Meta Level ◽  
Semantic Treatment ◽  
Completeness Theorems

In what follows there is presented a unified semantic treatment of certain “paradox-free” systems of entailment, including Church's weak theory of implication (Church [7]) and logics akin to the systems E and R of Anderson and Belnap (Anderson [3], Belnap [6]). We shall refer to these systems generally as relevant logics.The leading idea of the semantics is that just as in modal logic validity may be defined in terms of certain valuations on a binary relational structure so in relevant logics validity may be defined in terms of certain valuations on a semilattice—interpreted informally as the semilattice of possible pieces of information. Completeness theorems can be given relative to these semantics for the implicational fragments of relevant logics. The semantical viewpoint affords some insights into the structure of the systems—in particular light is thrown upon admissible modes of negation and on the assumptions underlying rejection of the “paradoxes of material implication”.The systems discussed are formulated in fragments of a first-order language with → (entailment), &, ⋁, ¬,(x) and (∃x) primitive, omitting identity but including a denumerable list of propositional variables (p, q, r, p1,…etc.), and (for each n > 0), a denumerable list of n-ary predicate letters. The schematic letters A, B, C, D, A1,… are used on the meta-level as variables ranging over formulas. The conventions of Church [9] are followed in abbreviating formulas. The semantics of the systems are given in informal terms; it is an easy matter to turn the informal descriptions into formal set-theoretical definitions.

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