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.

中文翻译：

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有限支撑结构的各种形式的无穷大

本文的目的是在有限支撑结构的框架中定义和研究无穷的概念，展示无穷基数的新属性。其中一些属性从非原子 Zermelo-Fraenkel 集合理论扩展到具有有限支持的原子对象世界，而其他属性则特定于有限支持的结构。我们比较了有限支持集合世界中无穷的替代定义，并提供了满足某些形式的无穷而不满足其他形式的原子集的相关示例。最后，我们提供了有限支持的可数集的表征。