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## Über dieses Buch

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.

## Inhaltsverzeichnis

### Chapter 1. Introduction

We start this chapter by presenting some motivation for using nominal sets and Fraenkel-Mostowski sets in the experimental sciences. We emphasize the subdivisions of the so-called Fraenkel-Mostowski framework by mentioning the Fraenkel-Mostowski permutation model of Zermelo-Fraenkel set theory with atoms, the Fraenkel-Mostowski axiomatic set theory, the theory of nominal sets, the theory of generalized nominal sets, and Extended Fraenkel-Mostowski set theory. Finally, we present an alternative mathematics for managing infinite structures in the experimental sciences. This is called Finitely Supported Mathematics and represents, informally, Zermelo-Fraenkel mathematics rephrased in terms of finitely supported structures.
Andrei Alexandru, Gabriel Ciobanu

### Chapter 2. Fraenkel-Mostowski Set Theory: A Framework for Finitely Supported Mathematics

In this chapter we present the basics of the Fraenkel-Mostowski framework, by studying concepts like invariant set, Fraenkel-Mostowski set, freshness quantifier, support, finiteness, fresh element, and abstraction. We also prove some original results regarding the consistency of various forms of choice in Finitely Supported Mathematics. Another goal of this chapter is to establish a connection between the theory of Fraenkel-Mostowski sets and the concept of logical notion presented by A. Tarski. More precisely, we prove that any invariant set from the Fraenkel-Mostowski universe is a logical notion in Tarski’s view. Moreover, the freshness quantifier is a logical symbol.
Andrei Alexandru, Gabriel Ciobanu

### Chapter 3. Algebraic Structures in Finitely Supported Mathematics

Finitely Supported Mathematics (FSM) is the mathematics developed in the framework of invariant/finitely supported structures. The aim of this chapter is to translate into FSM several algebraic concepts which were initially described using the Zermelo-Fraenkel axioms of set theory. We focus on multisets, generalized multisets, partially ordered sets and groups because these concepts are particularly relevant for experimental science. Moreover, we provide the main principles of translating a given algebraic concept into FSM. These principles are based on the remark that only finitely supported objects are allowed in FSM. We present in detail some FSM properties of the related algebraic structures, emphasizing the analogy between the results obtained in the framework of invariant sets and those obtained in the usual Zermelo-Fraenkel framework. This chapter may be read without using notions from higher-order logic, category theory, or the general equivariance principles for formulas in classical higher-order logic.
Andrei Alexandru, Gabriel Ciobanu

### Chapter 4. Extended Fraenkel-Mostowski Set Theory

The finite support axiom of Fraenkel-Mostowski set theory is very strong. We study the consequences of replacing this strong axiom with a weaker one. In this chapter we generalize Fraenkel-Mostowski set theory by giving a new set of axioms which defines Extended Fraenkel-Mostowski set theory. In Extended Fraenkel- Mostowski set theory, instead of the finite support axiom we require that each subset of the set of atoms is either finite or cofinite. We study several algebraic, order and topological properties of Extended Fraenkel-Mostowski sets.
Andrei Alexandru, Gabriel Ciobanu

### Chapter 5. Process Calculi in Finitely Supported Mathematics

The aim of this chapter is to present a set of compact transition rules (transition rules without side conditions) for the monadic version of the fusion calculus (update calculus). These transition rules are expressed using the quantifier ∀ and the freshness quantifier. Using some results presented in the second chapter of this book, we are able to compare the new semantics of the monadic fusion calculus with its related known semantics.
Andrei Alexandru, Gabriel Ciobanu

### Backmatter

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