Sets with atoms serve as an alternative to ZFC foundations for mathematics, where some infinite, though highly symmetric sets, behave in a finitistic way. Therefore, one can try to carry over analysis of the classical algorithms from finite structures to some infinite structures. Recent results show that this is indeed possible and leads to many practical applications. In this paper we shall take another route to finite analysis of infinite sets, which extends and sheds more light on sets with atoms. As an application of our theory we give a characterisation of languages recognized by automata definable in fragments of first-order logic.

中文翻译：

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超越原子集合：一阶逻辑中的可定义性

具有原子的集合可作为数学 ZFC 基础的替代，其中一些无限但高度对称的集合以有限的方式表现。因此，可以尝试将经典算法的分析从有限结构延续到一些无限结构。最近的结果表明，这确实是可能的，并导致了许​​多实际应用。在本文中，我们将采用另一条途径对无限集进行有限分析，它扩展并阐明了更多关于具有原子的集的信息。作为我们的理论的应用，我们给出了可在一阶逻辑片段中定义的自动机识别的语言的特征。