References in:
Timothy Williamson’s CoinFlipping Argument: Refuted Prior to Publication?
Erkenntnis 86 (3):575583 (2021)
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It is usual to identify initial conditions of classical dynamical systems with mathematical real numbers. However, almost all real numbers contain an infinite amount of information. I argue that a finite volume of space can’t contain more than a finite amount of information, hence that the mathematical real numbers are not physically relevant. Moreover, a better terminology for the socalled real numbers is “random numbers”, as their series of bits are truly random. I propose an alternative classical mechanics, which is (...) 

This article discusses how the concept of a fair finite lottery can best be extended to denumerably infinite lotteries. Techniques and ideas from nonstandard analysis are brought to bear on the problem. 



This is the second volume of philosophical essays by one of the most innovative and influential philosophers now writing in English. Containing thirteen papers in all, the book includes both new essays and previously published papers, some of them with extensive new postscripts reflecting Lewis's current thinking. The papers in Volume II focus on causation and several other closely related topics, including counterfactual and indicative conditionals, the direction of time, subjective and objective probability, causation, explanation, perception, free will, and rational (...) 



NonArchimedean probability functions allow us to combine regularity with perfect additivity. We discuss the philosophical motivation for a particular choice of axioms for a nonArchimedean 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 (...) 

The rejection of an infinitesimal solution to the zerofit problem by A. Elga ([2004]) does not seem to appreciate the opportunities provided by the use of internal finitelyadditive probability measures. Indeed, internal laws of probability can be used to find a satisfactory infinitesimal answer to many zerofit problems, not only to the one suggested by Elga, but also to the Markov chain (that is, discrete and memoryless) models of reality. Moreover, the generalization of likelihoods that Elga has in mind is (...) 

Isn't probability 1 certainty? If the probability is objective, so is the certainty: whatever has chance 1 of occurring is certain to occur. Equivalently, whatever has chance 0 of occurring is certain not to occur. If the probability is subjective, so is the certainty: if you give credence 1 to an event, you are certain that it will occur. Equivalently, if you give credence 0 to an event, you are certain that it will not occur. And so on for other (...) 

We prove, in ZFC,the existence of a definable, countably saturated elementary extension of the reals. 











Timothy Williamson has claimed to prove that regularity must fail even in a nonstandard setting, with a counterexample based on tossing a fair coin infinitely many times. I argue that Williamson’s argument is mistaken, and that a corrected version shows that it is not regularity which fails in the nonstandard setting but a fundamental property of shifts in Bernoulli processes. 

We prove, in ZFC, the existence of a definable, countably saturated elementary extension of the reals. 

