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001 978-3-319-42282-4
003 DE-He213
005 20200420221253.0
007 cr nn 008mamaa
008 160726s2016 gw | s |||| 0|eng d
020 _a9783319422824
_9978-3-319-42282-4
024 7 _a10.1007/978-3-319-42282-4
_2doi
050 4 _aQA75.5-76.95
072 7 _aUY
_2bicssc
072 7 _aUYA
_2bicssc
072 7 _aCOM014000
_2bisacsh
072 7 _aCOM031000
_2bisacsh
082 0 4 _a004.0151
_223
100 1 _aAlexandru, Andrei.
_eauthor.
245 1 0 _aFinitely Supported Mathematics
_h[electronic resource] :
_bAn Introduction /
_cby Andrei Alexandru, Gabriel Ciobanu.
264 1 _aCham :
_bSpringer International Publishing :
_bImprint: Springer,
_c2016.
300 _aVII, 185 p.
_bonline resource.
336 _atext
_btxt
_2rdacontent
337 _acomputer
_bc
_2rdamedia
338 _aonline resource
_bcr
_2rdacarrier
347 _atext file
_bPDF
_2rda
505 0 _aIntroduction -- 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 _aIn 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 _aComputer science.
650 0 _aComputers.
650 0 _aComputer science
_xMathematics.
650 0 _aAlgebra.
650 0 _aMathematical logic.
650 1 4 _aComputer Science.
650 2 4 _aTheory of Computation.
650 2 4 _aMathematics of Computing.
650 2 4 _aMathematical Logic and Foundations.
650 2 4 _aAlgebra.
700 1 _aCiobanu, Gabriel.
_eauthor.
710 2 _aSpringerLink (Online service)
773 0 _tSpringer eBooks
776 0 8 _iPrinted edition:
_z9783319422817
856 4 0 _uhttp://dx.doi.org/10.1007/978-3-319-42282-4
912 _aZDB-2-SCS
942 _cEBK
999 _c52712
_d52712