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 Introduction2 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 theory4 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 Pruss5 Dividends 5.1 Measure and utility 5.2 Regularity and uniformity 5.3 Credence and chance 5.4 Conditional probability6 General Considerations 6.1 Non-uniqueness 6.2 InvarianceAppendix .
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