Definability of models by means of existential formulas without identity

1993 ◽  
Vol 58 (2) ◽  
pp. 424-434 ◽  
Author(s):  
Paweł Pazdyka
Keyword(s):  
Binary Relation ◽  
Weak Point ◽  
Relation Symbol ◽  
Infinite Case ◽  
Binary Relation Symbol ◽  
Mathematical Literature ◽  
The Universe ◽  
Elementary Theories

The problem of coding relations by means of a single binary relation is well known in the mathematical literature. It was considered in interpretation theory, and also in connection with investigations of decidability of elementary theories. Using various constructions (see, e.g., [2,6], proofs of Theorem 11 in [7] and Theorem 16.51 in [3]), for any model for a countable language, one can construct a model for ℒp (a language with a single binary relation symbol ) in which is interpretable. Each of the mentioned constructions has the same weak point: the universe of is different than the universe of . In [4] we have shown that, in the infinite case, one can eliminate this defect, i.e., for any infinite , we have constructed a model , having the same universe as , in which is elementarily definable. In all constructions mentioned above, it appears that formulas, which define in ( in ), are very complicated. In the present paper, another construction of a model for ℒp is given.

On Ehrenfeucht-Fraïssé equivalence of linear orderings

1990 ◽  
Vol 55 (1) ◽  
pp. 65-73 ◽  
Author(s):  
Juha Oikkonen
Keyword(s):  
Binary Relation ◽  
Linear Ordering ◽  
Relation Symbol ◽  
Linear Orderings ◽  
Binary Relation Symbol

AbstractC. Karp has shown that if α is an ordinal with ωα = α and A is a linear ordering with a smallest element, then α and α ⊗ A are equivalent in L∞ω up to quantifer rank α. This result can be expressed in terms of Ehrenfeucht-Fraïssé games where player ∀ has to make additional moves by choosing elements of a descending sequence in α. Our aim in this paper is to prove a similar result for Ehrenfeucht-Fraïssé games of length ω1. One implication of such a result will be that a certain infinite quantifier language cannot say that a linear ordering has no descending ω1-sequences (when the alphabet contains only one binary relation symbol). Connected work is done by Hyttinen and Oikkonen in [H] and [O].

A note on countable complete theories having three isomorphism types of countable models

1976 ◽  
Vol 41 (3) ◽  
pp. 672-680 ◽  
Author(s):  
Robert E. Woodrow
Keyword(s):  
Binary Relation ◽  
Quantifier Elimination ◽  
Relation Symbol ◽  
Complete Theories ◽  
Binary Relation Symbol ◽  
Countable Models

AbstractWith quantifier elimination and restriction of language to a binary relation symbol and constant symbols it is shown that countable complete theories having three isomorphism types of countable models are “essentially” the Ehrenfeucht example [4, §6].

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”).

Provability with Finitely Many Variables

2002 ◽  
Vol 8 (3) ◽  
pp. 348-379 ◽  
Author(s):  
Robin Hirsch ◽  
Ian Hodkinson ◽  
Roger D. Maddux
Keyword(s):  
Predicate Logic ◽  
Algebraic Logic ◽  
Order Logic ◽  
First Order Logic ◽  
Alternative Semantics ◽  
Relation Symbol ◽  
First Order ◽  
Binary Predicate ◽  
Standard Set ◽  
Binary Relation Symbol

AbstractFor every finite n ≥ 4 there is a logically valid sentence φn with the following properties: φn contains only 3 variables (each of which occurs many times); φn contains exactly one nonlogical binary relation symbol (no function symbols, no constants, and no equality symbol); φn has a proof in first-order logic with equality that contains exactly n variables, but no proof containing only n − 1 variables. This result was first proved using the machinery of algebraic logic developed in several research monographs and papers. Here we replicate the result and its proof entirely within the realm of (elementary) first-order binary predicate logic with equality. We need the usual syntax, axioms, and rules of inference to show that φn has a proof with only n variables. To show that φn has no proof with only n − 1 variables we use alternative semantics in place of the usual, standard, set-theoretical semantics of first-order logic.

Rough approximations of bipolar soft sets by soft relations and their application in decision making

2021 ◽  
pp. 1-1
Author(s):  
Muhammad Shabir ◽  
Asad Mubarak ◽  
Munazza Naz
Keyword(s):  
Decision Making ◽  
Binary Relation ◽  
Rough Set Theory ◽  
Soft Set ◽  
Selection Problem ◽  
Soft Sets ◽  
Similarity Relations ◽  
Rough Approximation ◽  
Two Universes ◽  
The Universe

The rough set theory is an effective method for analyzing data vagueness, while bipolar soft sets can handle data ambiguity and bipolarity in many cases. In this article, we apply Pawlak’s concept of rough sets to the bipolar soft sets and introduce the rough bipolar soft sets by defining a rough approximation of a bipolar soft set in a generalized soft approximation space. We study their structural properties and discuss how the soft binary relation affects the rough approximations of a bipolar soft set. Two sorts of bipolar soft topologies induced by soft binary relation are examined. We additionally discuss some similarity relations between the bipolar soft sets, depending on their roughness. Such bipolar soft sets are very useful in the problems related to decision-making such as supplier selection problem, purchase problem, portfolio selection, site selection problem etc. A methodology has been introduced for this purpose and two algorithms are presented based upon the ongoing notions of foresets and aftersets respectively. These algorithms determine the best/worst choices by considering rough approximations over two universes i.e. the universe of objects and universe of parameters under a single framework of rough bipolar soft sets.

Structures in the Universe by Exact Methods

2009 ◽  
Author(s):  
Krzysztof Bolejko ◽  
Andrzej Krasinski ◽  
Charles Hellaby ◽  
Marie-Noelle Celerier
Keyword(s):  
Exact Methods ◽  
The Universe

The Riddle of the Universe at the Close of the Nineteenth Century

2009 ◽  
Author(s):  
Ernst Heinrich Philipp August Haeckel ◽  
Joseph McCabe
Keyword(s):  
Nineteenth Century ◽  
The Universe

The star that changed the universe

AccessScience ◽  
2015 ◽  
Keyword(s):  
The Universe

The shape of the universe

AccessScience ◽  
2015 ◽  
Keyword(s):  
The Universe
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