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Finitely Supported Mathematics - An Introduction (Paperback, Softcover reprint of the original 1st ed. 2016)
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Finitely Supported Mathematics - An Introduction (Paperback, Softcover reprint of the original 1st ed. 2016)
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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.
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