Infinitesimals hsmr12/28/2023 ![]() This approach really has never taken off, I think because when you include the. Bottom line: a very good way to understand the non-standard analysis approach, both for mathematicians and for interested students. ![]() 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. Keisler’s book is very good at the usage and applications of infinitesimals, particularly in setting up models of physical problems. 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. An infinitesimal object is an object less than any measurable size, but not so small or small that the available means cannot distinguish it from zero. Non-Archimedean probability functions allow us to combine regularity with perfect additivity. Infinitesimally small is a synonym of the word infinitesimal which means very, very small, or extremely small or vanishingly small or smaller than anything.
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