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Author Alexandru, Andrei

Title Finitely supported mathematics : an introduction / Andrei Alexandru, Gabriel Ciobanu
Published Cham : Springer, 2016
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Description 1 online resource (188 pages)
Contents 880-01 Acknowledgements; 1 Introduction; Abstract; 1.1 Motivation; 1.2 Approaches Related to the Fraenkel-Mostowski Framework; 1.3 Finitely Supported Mathematics; 1.4 Outline; 2 Fraenkel-Mostowski Set Theory: A Framework for Finitely Supported Mathematics; Abstract; 2.1 Axiom of Choice; 2.2 Permutative Renaming; 2.3 Sets with Atoms; 2.4 Constructions of Sets with Atoms; 2.4.1 Powersets; 2.4.2 Cartesian Products; 2.4.3 Disjoint Unions; 2.4.4 Function Spaces; 2.4.5 Categorical Constructions; 2.5 Fraenkel-Mostowski Axioms; 2.6 Logical Notions in the FM Cumulative Universe
880-01/(S Abstract5.1 Introduction and Methods; 5.2 A Case Study: The Monadic Fusion Calculus; 5.3 FSM Semantics of the Fusion Calculus; 5.4 FSM Semantics of Other Process Calculi; 5.4.1 FSM Semantics of the π-calculus; 5.4.1.1 Early Semantics of the π-calculus; 5.4.1.2 Late Semantics of the π-calculus; 5.4.1.3 FSM Semantics of the π-calculus; 5.4.2 FSM Semantics of the πI-calculus; 5.4.2.1 Sangiorgi Semantics of the πI-calculus; 5.4.2.2 FSM Semantics of the πI-calculus; 5.5 Comments; References
2.7 Inconsistency of Choice Principles2.8 Finiteness; 2.9 Freshness; 2.10 Abstraction; 2.10.1 Formal Definition; 2.10.2 Motivation; 2.10.3 Properties; 3 Algebraic Structures in Finitely Supported Mathematics; Abstract; 3.1 Multisets in Finitely Supported Mathematics; 3.1.1 Algebraic Properties of Multisets; 3.1.2 Multisets over Infinite Alphabets; 3.1.3 An Extension of the Framework; 3.2 Generalized Multisets in Finitely Supported Mathematics; 3.2.1 Algebraic Properties of Generalized Multisets; 3.2.1.1 Generalized Multisets as Groups; 3.2.1.2 Orders on Generalized Multisets
3.2.1.3 Generalized Multisets in Reverse Mathematics3.2.2 Generalized Multisets over Infinite Alphabets; 3.3 Order Theory in Finitely Supported Mathematics; 3.3.1 Partially Ordered Sets; 3.3.2 Galois Connections; 3.3.3 Rough Set Approximations; 3.3.4 Abstract Interpretation; 3.3.5 Calculability: Approximations of Fixed Points; 3.3.6 Complete Partially Ordered Sets; 3.3.7 Recursive Equations over CPOs; 3.4 Groups in Finitely Supported Mathematics; 3.4.1 Basic Results; 3.4.2 Isomorphism Theorems; 3.4.3 Embedding Theorems; 3.4.4 Finitely Supported Subgroups; 3.5 General Comments
3.6 Comments on the Methods Used in This Chapter3.7 Conclusion; 4 Extended Fraenkel-Mostowski Set Theory; Abstract; 4.1 Axioms of Extended Fraenkel-Mostowski Set Theory; 4.2 Inconsistency of the Axiom of Choice; 4.3 Algebraic Properties of EFM Sets; 4.4 Topological Properties of EFM Sets; 4.4.1 Subgroup Lattices as Domains; 4.4.2 Scott Topology over the Subgroup Lattice of a Group; 4.4.3 Topological Properties of the Group of Permutations of Atoms in EFM Set Theory; 4.5 Renamings in the Extended Fraenkel-Mostowski Framework; 4.6 Comments; 5 Process Calculi in Finitely Supported Mathematics
Summary 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
Bibliography Includes bibliographical references
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Subject Set theory.
Mathematics.
Form Electronic book
Author Ciobanu, Gabriel.
ISBN 9783319422824 (electronic bk.)
3319422820 (electronic bk.)
3319422812 (print)
9783319422817 (print)
(print)