# Predicting the future of arbitrary functions

Let $$S$$ be any non-empty set. If you have a function $$f\colon \mathbb{R} \to S$$ and you tell me its values on an interval $$(-\infty,t)$$, can I predict which value it will have at the time $$t$$? If the function is continuous, then of course I can; and in general not. Now interestingly, if you ask not me but Hardin and Taylor, then they can predict $$f(t)$$ for almost all $$t$$ knowing only the values of $$f$$ before the time $$t$$!

Let me make their result explicit. Denote by $$S^\mathbb{R}_\bullet$$ the set of all functions $$f\colon (-\infty,t_f) \to S$$; note that the domain $$(-\infty,t_f)$$ depends on the function $$f$$. We call any mapping $$P\colon S^\mathbb{R}_\bullet \to S$$ an $$S$$-predictor: if we throw any function $$f\colon (-\infty,t_f) \to S$$ into $$P$$, then the result $$P(f)$$ is the attempt to predict which value $$f$$ will take at the time $$t_f \in \mathbb{R}$$.

We can now ask for which functions $$F\colon \mathbb{R} \to S$$ and which times $$t \in \mathbb{R}$$ the equality $P(F|_{(-\infty,t)}) = F(t)$ hold true. The result of Hardin and Taylor is that for every set $$S$$ there exists an $$S$$-predictor for which this equality holds true for all functions $$F\colon \mathbb{R} \to S$$ for almost all times $$t \in \mathbb{R}$$ (where the measure-zero set of `bad’ predictions depends on the function $$F$$).

Reference: Hardin-Taylor, A Peculiar Connection between the Axiom of Choice and Predicting the Future, The American Mathematical Monthly vol. 115 (2008).

Posted on