# Infinities within Finitely Supported Structures

## TL;DR

This paper explores the concept of infinity within finitely supported algebraic structures, extending classical set theory notions to atomic objects with finite support, and characterizes finitely supported countable sets.

## Contribution

It introduces and analyzes different types of infinities and cardinalities in finitely supported structures, extending classical properties and providing new characterizations.

## Key findings

- Extended properties of infinite cardinalities to finitely supported structures
- Compared alternative definitions of infinite finitely supported sets
- Characterized finitely supported countable sets

## Abstract

The theory of finitely supported algebraic structures is related to Pitts theory of nominal sets (by equipping finitely supported sets with finitely supported internal algebraic laws). It represents a reformulation of Zermelo Fraenkel set theory obtained by requiring every set theoretical construction to be finitely supported according to a certain action of a group of permutations of some basic elements named atoms. Its main purpose is to let us characterize infinite algebraic structures, defined involving atoms, only by analyzing their finite supports. The first goal of this paper is to define and study different kinds of infinities and the notion of `cardinality' in the framework of finitely supported structures. We present several properties of infinite cardinalities. Some of these properties are extended from the non-atomic Zermelo Fraenkel set theory into the world of atomic objects with finite support, while other properties are specific to finitely supported structures. We also compare alternative definitions of `infinite finitely supported set', and we finally provide a characterization of finitely supported countable sets.