Free Chances Are by Michael Kaplan

system glided over the crack between stating equally possible outcomes and assuming them in experiment, von Mises’ straddled a crevasse when it assumed that the relative frequencies of observed events could indeed approach a limiting value. All might be revealed in the long run—but, as Keynes pointed out, in the long run we are all dead.
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    Trains of thought can sometimes seem exactly that; bright little worlds rattling through unknown and perhaps uninhabited darkness. It is the cliché tragedy of intellectual life to discover, far too late, that your own train got switched off the main line up a remote spur whose rusty rails and lumpy roadbed reveal all too clearly that it leads only to ghost towns. Sometimes, though, there is the excitement of approaching the metropolis on converging lines: other golden windows glide alongside with other travelers folding their newspapers or reaching for their coats, other children waving in welcome.
    Lebesgue’s measure theory and von Mises’ idea of frequency were ideas that were already within view of one another. Between them, two further trains were advancing: one was the growing conviction among physicists that certain processes in thermodynamics and quantum mechanics had only a probabilistic meaning—that there was no mechanical, deterministic model by which they could be described or even imagined. The other was the great contemporary movement, which promised that of all of mathematics—all the rigor and complexity that, magically, seem to find so many parallels with the richness and beauty of life—could be founded on a few axioms linked by the rules of deductive logic. There was a palpable sense of approaching the terminus, where the many travelers could exchange stories—all in the same language.
    This shared language was the notion of the set, introduced by Georg Cantor as a means of keeping the infinite in mind without having to think of infinite things or infinite processes. The definition of a set is intentionally as loose as possible: it is defined by its members. But membership can be generated by the most varied of rules: “numbers divisible by 5” determines a set; but so does “Dog; 17; Red.” It is possible to have an empty set: we can talk about the contents of the box even though there’s nothing in it. We can subdivide a set into subsets, which will also be sets. We can combine sets into a union, which is also a set. We can define where two sets overlap; the overlap or intersection is also a set. We can have infinite sets (such as all the counting numbers). And we can imagine the collection of all subsets of a set—which is also a set.
    What, you might feel yourself asking, are we actually talking about here? Nothing in particular—and that’s the point. This is pure form; its aim is to support a logical system that governs every instance of the way we consider some things as distinct from other things. A set is simply a pair of mental brackets, isolating “this” from “not this”; we can put into those brackets whatever interests us. Just as Cantor’s infinity can be put in a set and considered here and now, rather than endlessly and forever, von Mises’ indefinite sequences of observations can constitute a set: the set of events of observing something. We can use the axioms by which we manipulate sets to manipulate collections of events. Most important, Lebesgue’s concept of measure gives us a method for assigning a unique and complete value to a set, its subsets, and its elements—and these values behave the way we want probability values to behave.
    Lines of thought, all coming together, all converging—so who better to effect the final union than a man who was born on a train? Andrei Nikolaevich Kolmogorov was a son any parent would be proud of, but neither parent ever saw him. His mother died in bearing him, on April 25, 1903, journeying from the Crimea