About this book
Introduction
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 FraenkelMostowski (FM) permutative model of ZermeloFraenkel (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 FraenkelMostowski (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.
Keywords
Bibliographic information
 Book Title Finitely Supported Mathematics
 Book Subtitle An Introduction

Authors
Andrei Alexandru
Gabriel Ciobanu
 DOI https://doi.org/10.1007/9783319422824
 Copyright Information Springer International Publishing Switzerland 2016
 Publisher Name Springer, Cham
 eBook Packages Computer Science Computer Science (R0)
 Hardcover ISBN 9783319422817
 Softcover ISBN 9783319825458
 eBook ISBN 9783319422824
 Edition Number 1
 Number of Pages VII, 185
 Number of Illustrations 0 b/w illustrations, 0 illustrations in colour

Topics
Theory of Computation
Mathematics of Computing
Mathematical Logic and Foundations
Algebra
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