Non-Archimedean probability functions allow us to combine regularity with perfect additivity. We discuss the philosophical motivation for a particular choice of axioms for a non-Archimedean probability theory and answer some philosophical objections that have been raised against infinitesimal probabilities in general. 1 Introduction 2 The Limits of Classical Probability Theory 2.1 Classical probability functions 2.2 Limitations 2.3 Infinitesimals to the rescue? 3 NAP Theory 3.1 First four axioms of NAP 3.2 Continuity and conditional probability 3.3 The final axiom of NAP 3.4 Infinite sums 3.5 Definition of NAP functions via infinite sums 3.6 Relation to numerosity theory 4 Objections and Replies 4.1 Cantor and the Archimedean property 4.2 Ticket missing from an infinite lottery 4.3 Williamson’s infinite sequence of coin tosses 4.4 Point sets on a circle 4.5 Easwaran and Pruss 5 Dividends 5.1 Measure and utility 5.2 Regularity and uniformity 5.3 Credence and chance 5.4 Conditional probability 6 General Considerations 6.1 Non-uniqueness 6.2 Invariance Appendix 1 Introduction 2 The Limits of Classical Probability Theory 2.1 Classical probability functions 2.2 Limitations 2.3 Infinitesimals to the rescue? 2.1 Classical probability functions 2.2 Limitations 2.3 Infinitesimals to the rescue? 3 NAP Theory 3.1 First four axioms of NAP 3.2 Continuity and conditional probability 3.3 The final axiom of NAP 3.4 Infinite sums 3.5 Definition of NAP functions via infinite sums 3.6 Relation to numerosity theory 3.1 First four axioms of NAP 3.2 Continuity and conditional probability 3.3 The final axiom of NAP 3.4 Infinite sums 3.5 Definition of NAP functions via infinite sums 3.6 Relation to numerosity theory 4 Objections and Replies 4.1 Cantor and the Archimedean property 4.2 Ticket missing from an infinite lottery 4.3 Williamson’s infinite sequence of coin tosses 4.4 Point sets on a circle 4.5 Easwaran and Pruss 4.1 Cantor and the Archimedean property 4.2 Ticket missing from an infinite lottery 4.3 Williamson’s infinite sequence of coin tosses 4.4 Point sets on a circle 4.5 Easwaran and Pruss 5 Dividends 5.1 Measure and utility 5.2 Regularity and uniformity 5.3 Credence and chance 5.4 Conditional probability 5.1 Measure and utility 5.2 Regularity and uniformity 5.3 Credence and chance 5.4 Conditional probability 6 General Considerations 6.1 Non-uniqueness 6.2 Invariance 6.1 Non-uniqueness 6.2 Invariance Appendix 

中文翻译：

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无穷小概率：表 1。

非阿基米德概率函数允许我们将正则性与完美可加性结合起来。我们讨论了非阿基米德概率论公理的特定选择的哲学动机，并回答了一些针对一般无穷小概率提出的哲学反对意见。1 引言 2 经典概率论的局限性 2.1 经典概率函数 2.2 局限性 2.3 拯救无穷小？3 NAP 理论 3.1 NAP 的前四个公理 3.2 连续性和条件概率 3.3 NAP 的最后一个公理 3.4 无限和 3.5 通过无限和定义 NAP 函数 3.6 与数论的关系 4 反对和答复 4.1 康托尔财产和 Archim4.1无限彩票 4.3 威廉姆森的无限掷硬币序列 4. 4 圆上的点集 4.5 Easwaran 和 Pruss 5 红利 5.1 测度和效用 5.2 规律性和一致性 5.3 可信度和机会 5.4 条件概率 6 一般考虑 6.1 非唯一性 6.2 不变性 附录 1 引言 2 经典概率论的局限性 1 概率理论函数2.2 限制 2.3 救命的无穷小？2.1 经典概率函数 2.2 局限性 2.3 拯救无穷小？3 NAP 理论 3.1 NAP 的前四个公理 3.2 连续性和条件概率 3.3 NAP 的最后一个公理 3.4 无穷和 3.5 通过无穷和定义 NAP 函数 3.6 与数论的关系 3.1 NAP 的前四个公理 最终连续性 3.32 条件和概率NAP 公理 3.4 无限和 3.5 通过无限和 3 定义 NAP 函数。