Infinitesimal Probabilities

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 .