Does the Hardin-Taylor Predictor Actually Predict the Future? Yes; but Only in Retrospect

Abstract

The Hardin-Taylor theorem, a consequence of the axiom of choice, gives a strategy to predict future values of an arbitrary function of time $f:\mathbb{R}\to S$ given its past values. The times in $\mathbb{R}$ at which the strategy fails are shown to form a countable nowhere dense set for any function $f$. The theorem is often presented as a striking result about predicting the future. In this note we discuss the operational content of the theorem and clarify this frequent interpretation. The Hardin--Taylor prediction strategy is online: at time $t$ it depends only on the observed past $f \upharpoonright (-\infty,t)$. Its performance guarantee, however, is offline: the exceptional set consisting of times at which the predictor fails is defined only after the entire function has been realized. We spell out the precise definition of the predictor, discuss a simple two-function example as a sanity check, and emphasize the retrospective nature of verifying the predictor's success. We also examine the proper probabilistic setting in which the success likelihood of the predictor as a forecasting procedure should be evaluated. Our conclusion is that the theorem is best understood as a result about the smallness of the set of times at which the predictor must be course-corrected, rather than as a prospectively verifiable procedure for forecasting the future.