A Peculiar Connection Between the Axiom of Choice and Predicting the Future

Abstract

INTRODUCTION.We often model systems that change over time as functions from the real numbers R (or a subinterval of R) into some set S of states, and it is often our goal to predict the behavior of these systems. Generally, this requires rules governing their behavior, such as a set of differential equations or the assumption that the system (as a function) is analytic. With no such assumptions, the system could be an arbitrary function, and the values of arbitrary functions are notoriously hard to predict. After all, if someone proposed a strategy for predicting the values of an arbitrary function based on its past values, a reasonable response might be, "That is impossible. Given any strategy for predicting the values of an arbitrary function, one could just define a function that diagonalizes against it: whatever the strategy predicts, define the function to be something else." This argument, however, makes an appeal to induction: to diagonalize against the proposed strategy at a point t, we must have already defined our function for all s < t in order to determine what the strategy would predict at t. So, the argument would only be valid if R were well-ordered, but R is emphatically not well-ordered.In fact, the lack of well-orderedness in the reals can be exploited to produce a very counterintuitive result: there is a strategy for predicting the values of an arbitrary function, based on its previous values, that is almost always correct. Specifically, given the values of a function on an interval (−∞, t), the strategy produces a guess for the values of the function on [t, ∞), and at all but countably many t, there is an ε > 0 such that the prediction is valid on [t, t + ε). Noting that any countable set of reals has measure 0, we can restate this informally: at almost every instant t, the strategy predicts some "ε-glimpse" of the future.We should emphasize that 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. For instance, "predicting the present" is a very natural way to think of the problem of guessing the value of f (t) based on f |(−∞, t).

Anonymity in Predicting the Future

“…The details of the proof can be found in [1], but the idea is simple. Using the axiom of choice, we can let ≺ be a well ordering of R S. Now for any f ∈ R S and any…”

“…Proof. The predictor we use is similar to the one described in [1]. Using the axiom of choice, fix a well ordering ≺ of R S that lists periodic scenarios first, then other affine-invariant scenarios, and then the rest of the scenarios.…”

“…In [1], Christopher Hardin and Alan Taylor present a remarkable consequence of the axiom of choice. Consider an arbitrary set S and a function f : R → S. We think of the domain of f as representing time, and for each x ∈ R, we think of f (x) as the state of some system at time x.…”

The mathematics of coordinated inference: a study of generalized hat problems

“…The second result we derive concerns the extent to which "the present can be predicted based on the past." Here, the exact characterization of the error sets occurs in Theorems 3.1 and 3.5 in [HT08b].…”

The role of measurability in game-theoretic probability

“…These authors show in [4] (see also [5,Sect. 7.4]) that the axiom of choice provides us with an Occam-type strategy able to predict the short-term future (usually albeit not always), which makes it easy to get rich when allowed to trade in a security whose price changes in a non-trivial manner.…”