  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.
