Christian Espindola: How to predict the future with the axiom of choice

Generally we model systems that change over time by using functions defined on the real numbers (or on some subset of them) and that take values in a certain set of states S, and we often try to predict how such systems will behave. Although generally this requires some rules governing the system (as, for example, some set of differential equations), when no hypothesis is made, the resulting function is completely arbitrary and quite difficult to predict.

In a recent work (2008) of Christopher Hardin and Alan Taylor published in the Monthly, the authors show how one can use the well-ordering theorem to produce a counter-intuitive result: there is a strategy that allows to find out the values of an arbitrary function in a given instant knowing its previous values, and that is almost always correct. More precisely: given the values of a function in (-∞,t), there is a strategy that allows to predict the values in the interval [t,t+ε) for some positive \epsilon depending on t, in such a way that, except for a countable number of values of t, the prediction is correct. In other words, for almost every t such strategy “opens an ε-window to the future”.

In this talk we will discuss this result, its proof and its consequences. To avoid sensationalism we cite, nevertheless, the authors own words: “these results do not give a practical means of predicting the future, just as the time dilation one would experience standing near the event horizon of a black hole does not give a practical time machine. Nevertheless, we choose this presentation because we find it the most interesting, as well as pedagogically useful”.