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).