The Game of Infinity Questions
Channel: John Mount
Date: 2021-02-14

I use "the game of infinity questions" to motivate some of the common axioms of measure theory. The idea is: the axioms about infinite intersections are designed to prevent a certain type of lying from hiding in distributions.

I have a new math chalk talk up: The Game of Infinity Questions. https://win-vector.com/2021/02/14/new-math-chalk-talk-the-game-of-infinity-questions/ 

This is back to establishing the "reasonableness" of Kolmogorov's Axiom of continuity (in his actual formulation of his axiomatization of probability https://win-vector.com/2020/09/19/kolmogorovs-axioms-of-probability-even-smarter-than-you-have-been-told/  ).

Remember, his argument is "it is a bit off to have strong opinions on infinite processes, as we will never live to see one." So he argues, "let's pick axioms that make infinite processes look a lot like finite ones, as you will never know the difference." This can fail. Not all things that are true in the finite case can hold in the infinite case, we get contridictions if we so try. So what one picks as the "obvious properties to keep" completely changes what properties we are ascribing to infinite sets and infinite processes. So the games is: show try to show the alternative to what you want looks awful.

In the game of infinity questions, we try to eliminate probability distritutions where limits are not respected.