國立高雄大學圖資館 |

Language: English

Back

Finitely supported mathematicsan int...

Alexandru, Andrei.

Finitely supported mathematicsan introduction /

Record Type:	Electronic resources : Monograph/item
Title/Author:	Finitely supported mathematicsby Andrei Alexandru, Gabriel Ciobanu.
Reminder of title:	an introduction /
Author:	Alexandru, Andrei.
other author:	Ciobanu, Gabriel.
Published:	Cham :Springer International Publishing :2016.
Description:	vii, 185 p. :ill., digital ;24 cm.
Contained By:	Springer eBooks
Subject:	Mathematics.
Online resource:	http://dx.doi.org/10.1007/978-3-319-42282-4
ISBN:	9783319422824$q(electronic bk.)

Finitely supported mathematicsan introduction /
Alexandru, Andrei.

Finitely supported mathematicsan introduction /[electronic resource] :by Andrei Alexandru, Gabriel Ciobanu. - Cham :Springer International Publishing :2016. - vii, 185 p. :ill., digital ;24 cm.

Introduction -- Fraenkel-Mostowski Set Theory: A Framework for Finitely Supported Mathematics -- Algebraic Structures in Finitely Supported Mathematics -- Extended Fraenkel-Mostowski Set Theory -- Process Calculi in Finitely Supported Mathematics -- References.

In this book the authors present an alternative set theory dealing with a more relaxed notion of infiniteness, called finitely supported mathematics (FSM) It has strong connections to the Fraenkel-Mostowski (FM) permutative model of Zermelo-Fraenkel (ZF) set theory with atoms and to the theory of (generalized) nominal sets. More exactly, FSM is ZF mathematics rephrased in terms of finitely supported structures, where the set of atoms is infinite (not necessarily countable as for nominal sets) In FSM, 'sets' are replaced either by `invariant sets' (sets endowed with some group actions satisfying a finite support requirement) or by `finitely supported sets' (finitely supported elements in the powerset of an invariant set) It is a theory of `invariant algebraic structures' in which infinite algebraic structures are characterized by using their finite supports. After explaining the motivation for using invariant sets in the experimental sciences as well as the connections with the nominal approach, admissible sets and Gandy machines (Chapter 1), the authors present in Chapter 2 the basics of invariant sets and show that the principles of constructing FSM have historical roots both in the definition of Tarski `logical notions' and in the Erlangen Program of Klein for the classification of various geometries according to invariants under suitable groups of transformations. Furthermore, the consistency of various choice principles is analyzed in FSM. Chapter 3 examines whether it is possible to obtain valid results by replacing the notion of infinite sets with the notion of invariant sets in the classical ZF results. The authors present techniques for reformulating ZF properties of algebraic structures in FSM. In Chapter 4 they generalize FM set theory by providing a new set of axioms inspired by the theory of amorphous sets, and so defining the extended Fraenkel-Mostowski (EFM) set theory. In Chapter 5 they define FSM semantics for certain process calculi (e.g., fusion calculus), and emphasize the links to the nominal techniques used in computer science. They demonstrate a complete equivalence between the new FSM semantics (defined by using binding operators instead of side conditions for presenting the transition rules) and the known semantics of these process calculi. The book is useful for researchers and graduate students in computer science and mathematics, particularly those engaged with logic and set theory.

ISBN: 9783319422824$q(electronic bk.)

Standard No.: 10.1007/978-3-319-42282-4doiSubjects--Topical Terms:

184409
Mathematics.

LC Class. No.: QA248 / .A44 2016

Dewey Class. No.: 511.322

Finitely supported mathematicsan introduction /
LDR:03722nmm a2200337 a 4500 001 492788
003 DE-He213
005 20161215144806.0
006 m d
007 cr nn 008maaau
008 170220s2016 gw s 0 eng d
020 $a 9783319422824$q(electronic bk.)
020 $a 9783319422817$q(paper)
024 7 $a 10.1007/978-3-319-42282-4 $2 doi
035 $a 978-3-319-42282-4
040 $a GP $c GP
041 0 $a eng
050 4 $a QA248 $b .A44 2016
072 7 $a UY $2 bicssc
072 7 $a UYA $2 bicssc
072 7 $a COM014000 $2 bisacsh
072 7 $a COM031000 $2 bisacsh
082 0 4 $a 511.322 $2 23
090 $a QA248 $b .A383 2016
100 1 $a Alexandru, Andrei. $3 753194
245 1 0 $a Finitely supported mathematics $h [electronic resource] : $b an introduction / $c by Andrei Alexandru, Gabriel Ciobanu.
260 $a Cham : $b Springer International Publishing : $b Imprint: Springer, $c 2016.
300 $a vii, 185 p. : $b ill., digital ; $c 24 cm.
505 0 $a Introduction -- Fraenkel-Mostowski Set Theory: A Framework for Finitely Supported Mathematics -- Algebraic Structures in Finitely Supported Mathematics -- Extended Fraenkel-Mostowski Set Theory -- Process Calculi in Finitely Supported Mathematics -- References.
520 $a In this book the authors present an alternative set theory dealing with a more relaxed notion of infiniteness, called finitely supported mathematics (FSM) It has strong connections to the Fraenkel-Mostowski (FM) permutative model of Zermelo-Fraenkel (ZF) set theory with atoms and to the theory of (generalized) nominal sets. More exactly, FSM is ZF mathematics rephrased in terms of finitely supported structures, where the set of atoms is infinite (not necessarily countable as for nominal sets) In FSM, 'sets' are replaced either by `invariant sets' (sets endowed with some group actions satisfying a finite support requirement) or by `finitely supported sets' (finitely supported elements in the powerset of an invariant set) It is a theory of `invariant algebraic structures' in which infinite algebraic structures are characterized by using their finite supports. After explaining the motivation for using invariant sets in the experimental sciences as well as the connections with the nominal approach, admissible sets and Gandy machines (Chapter 1), the authors present in Chapter 2 the basics of invariant sets and show that the principles of constructing FSM have historical roots both in the definition of Tarski `logical notions' and in the Erlangen Program of Klein for the classification of various geometries according to invariants under suitable groups of transformations. Furthermore, the consistency of various choice principles is analyzed in FSM. Chapter 3 examines whether it is possible to obtain valid results by replacing the notion of infinite sets with the notion of invariant sets in the classical ZF results. The authors present techniques for reformulating ZF properties of algebraic structures in FSM. In Chapter 4 they generalize FM set theory by providing a new set of axioms inspired by the theory of amorphous sets, and so defining the extended Fraenkel-Mostowski (EFM) set theory. In Chapter 5 they define FSM semantics for certain process calculi (e.g., fusion calculus), and emphasize the links to the nominal techniques used in computer science. They demonstrate a complete equivalence between the new FSM semantics (defined by using binding operators instead of side conditions for presenting the transition rules) and the known semantics of these process calculi. The book is useful for researchers and graduate students in computer science and mathematics, particularly those engaged with logic and set theory.
650 0 $a Mathematics. $3 184409
650 0 $a Set theory. $3 190892
650 1 4 $a Computer Science. $3 212513
650 2 4 $a Theory of Computation. $3 274475
650 2 4 $a Mathematics of Computing. $3 273710
650 2 4 $a Mathematical Logic and Foundations. $3 274479
650 2 4 $a Algebra. $3 188312
700 1 $a Ciobanu, Gabriel. $3 250690
710 2 $a SpringerLink (Online service) $3 273601
773 0 $t Springer eBooks
856 4 0 $u http://dx.doi.org/10.1007/978-3-319-42282-4
950 $a Computer Science (Springer-11645)