scholarly journals Spectra and satisfiability for logics with successor and a unary function

2018 ◽  
Vol 64 (4-5) ◽  
pp. 286-311
Author(s):  
Arthur Milchior
Keyword(s):  
Unary Function

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.

Uniform model-completeness for the real field expanded by power functions

2010 ◽  
Vol 75 (4) ◽  
pp. 1441-1461
Author(s):  
Tom Foster
Keyword(s):  
Real Field ◽  
Power Functions ◽  
Model Completeness ◽  
The Real ◽  
Unary Function ◽  
First Order ◽  
Existential Formula ◽  
Uniform Model

AbstractWe prove that given any first order formula ϕ in the language L′ = {+, ·, <,(fi)iЄI, (ci)iЄI}, where the fi are unary function symbols and the ci are constants, one can find an existential formula Ψ such that φ and Ψ are equivalent in any L′-structure

Pfaffian differential equations over exponential o-minimal structures

2002 ◽  
Vol 67 (1) ◽  
pp. 438-448 ◽  
Author(s):  
Chris Miller ◽  
Patrick Speissegger
Keyword(s):  
Asymptotic Behavior ◽  
Differential Equations ◽  
Asymptotic Integration ◽  
Real Field ◽  
Asymptotic Behavior Of Solutions ◽  
The Real ◽  
Unary Function ◽  
Ordered Fields ◽  
Component Function

In this paper, we continue investigations into the asymptotic behavior of solutions of differential equations over o-minimal structures.Let ℜ be an expansion of the real field (ℝ, +, ·).A differentiable map F = (F1,…, F1): (a, b) → ℝi is ℜ-Pfaffian if there exists G: ℝ1+l → ℝl definable in ℜ such that F′(t) = G(t, F(t)) for all t ∈ (a, b) and each component function Gi: ℝ1+l → ℝ is independent of the last l − i variables (i = 1, …, l). If ℜ is o-minimal and F: (a, b) → ℝl is ℜ-Pfaffian, then (ℜ, F) is o-minimal (Proposition 7). We say that F: ℝ → ℝl is ultimately ℜ-Pfaffian if there exists r ∈ ℝ such that the restriction F ↾(r, ∞) is ℜ-Pfaffian. (In general, ultimately abbreviates “for all sufficiently large positive arguments”.)The structure ℜ is closed under asymptotic integration if for each ultimately non-zero unary (that is, ℝ → ℝ) function f definable in ℜ there is an ultimately differentiable unary function g definable in ℜ such that limt→+∞[g′(t)/f(t)] = 1- If ℜ is closed under asymptotic integration, then ℜ is o-minimal and defines ex: ℝ → ℝ (Proposition 2).Note that the above definitions make sense for expansions of arbitrary ordered fields.

On hyper-torre isols

1989 ◽  
Vol 54 (4) ◽  
pp. 1160-1166 ◽  
Author(s):  
Rod Downey
Keyword(s):  
Natural Numbers ◽  
Unary Function

As Dekker [3] suggested, certain fragments of the isols can exhibit an arithmetic rather more resembling that of the natural numbers than the general isols do. One such natural fragment is Barback's “tame models” (cf. [2], [6] and [7]), whose roots go back to Nerode [8]. In this paper we study another variety of such fragments: the hyper-torre isols introduced by Ellentuck [4]. Let Y denote an infinite isol with D(Y) the collection of all isols A ≤ f∧(Y) for some recursive and combinational unary function f. (Here, as usual, f∧ is the Myhill-Nerode extension of f to the isols).

scholarly journals Completeness of two theories on ordered abelian groups and embedding relations

1980 ◽  
Vol 77 ◽  
pp. 33-39 ◽  
Author(s):  
Yuichi Komori
Keyword(s):  
Binary Relation ◽  
Abelian Groups ◽  
Function Symbol ◽  
Binary Function ◽  
Relation Symbol ◽  
Unary Function ◽  
First Order ◽  
Order Language ◽  
Unary Relation ◽  
Binary Relation Symbol

The first order language ℒ that we consider has two nullary function symbols 0, 1, a unary function symbol –, a binary function symbol +, a unary relation symbol 0 <, and the binary relation symbol = (equality). Let ℒ′ be the language obtained from ℒ, by adding, for each integer n > 0, the unary relation symbol n| (read “n divides”).

Counting finite models

1997 ◽  
Vol 62 (3) ◽  
pp. 925-949 ◽  
Author(s):  
Alan R. Woods
Keyword(s):  
Additional Condition ◽  
Fixed Number ◽  
Second Order ◽  
Single Component ◽  
Partial Function ◽  
Unary Function ◽  
Finite Models ◽  
Finite Structure ◽  
Generating Series

AbstractLetφbe a monadic second order sentence about a finite structure from a classwhich is closed under disjoint unions and has components. Compton has conjectured that if the number ofnelement structures has appropriate asymptotics, then unlabelled (labelled) asymptotic probabilitiesν(φ)(μ(φ)respectively) forφalways exist. By applying generating series methods to count finite models, and a tailor made Tauberian lemma, this conjecture is proved under a mild additional condition on the asymptotics of the number of single component-structures. Prominent among examples covered, are structures consisting of a single unary function (or partial function) and a fixed number of unary predicates.

Compactness of a supervaluational language

1983 ◽  
Vol 48 (2) ◽  
pp. 384-386 ◽  
Author(s):  
E. Bencivenga
Keyword(s):  
Open Problem ◽  
Present Note ◽  
Complete Solution ◽  
The Other ◽  
Unary Function ◽  
Power Set ◽  
Binary Predicate

Compactness of supervaluational semantics has long been an open problem. Recently, Peter Woodruff [2] showed that the full quantificational language is not compact. In the present note, I will show that the quantifier-free fragment of the language is compact. Since this result can be easily extended to the monadic quantificational language, and since Woodruff's result only requires the presence of one binary predicate, the two results combined give a complete solution of the problem.My language L contains infinitely many individual constants, infinitely many n-ary predicates (for every n > 0), the usual connectives, the symbols E! and =, and parentheses. (Atomic) sentences are defined as usual, and Ab/a is the result of uniformly substituting b for a in the sentence A.An interpretation of L is an ordered pair I = 〈D, φ〉, where D is a set (possibly empty) and φ is a unary function, total from the set of n-ary predicates into the power set of Dn, and partial from the set of constants into D.A classical valuation for an interpretation I = 〈D, φ〉 is a total unary function V from the set of sentences into {T, F} such that(a) if φ is defined for all of a1, …, am, then V(Pa1 … an) = T iff 〈φ(a1), …, φ(an)〉 ∈ φ(P);(b) if φ(a) and φ(b) are both defined, then V(a = b) = T iff φ(a) = φ(b;(c) if exactly one of φ(a) and φ(b) is defined, then V(a = b) = F;(d) V(a = a) = T;(e) if V(a = b) = T and A is atomic, then V(A) = V(Ab/a);(f)V(E!a) = T iff φ(a) is defined;(g) V(~ A) = T iff V(A) = F, and similarly for the other connectives.

A monotonicity theorem for dp-minimal densely ordered groups

2010 ◽  
Vol 75 (1) ◽  
pp. 221-238 ◽  
Author(s):  
John Goodrick
Keyword(s):  
General Notion ◽  
Ordered Group ◽  
Common Generalization ◽  
Unary Function ◽  
Monotonic Functions ◽  
Minimal Theory ◽  
Ordered Groups ◽  
Weak Minimality

AbstractDp-minimality is a common generalization of weak minimality and weak o-minimality. If T is a weakly o-minimal theory then it is dp-minimal (Fact 2.2), but there are dp-minimal densely ordered groups that are not weakly o-minimal. We introduce the even more general notion of inp-minimality and prove that in an inp-minimal densely ordered group, every definable unary function is a union of finitely many continuous locally monotonic functions (Theorem 3.2).

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