The goal of this paper is to define and study the notion of infinity in the framework of finitely supported structures, presenting new properties of infinite cardinalities. Some of these properties are extended from the non-atomic Zermelo–Fraenkel set theory to the world of atomic objects with finite support, while other properties are specific to finitely supported structures. We compare alternative definitions for infinity in the world of finitely supported sets, and provide relevant examples of atomic sets which satisfy some forms (...) of infinity, while do not satisfy others. Finally, we provide a characterization of finitely supported countable sets. (shrink)