In my previous post, I showed that given a backwards infinite sequence of coin tosses, there is a simple strategy leveraging data about infinitely many past coin flips that guarantees that you guessed correctly infinitely often. I then suggested that this supports the idea that one can't leverage an infinite amount of past data, and that in turn supports causal finitism--the denial of the possibility of infinite causal histories. But there is a gap in that argument: Maybe there is some strategy that guarantees infinitely many correct guesses that doesn't require the guesser to make use of data about infinitely many past coin flips. If so, then the paradox doesn't have much to do with infinite amounts of data.
Fortunately for me, that gap can be filled (modulo the Axiom of Choice). Given the Axiom of Choice, it's a theorem that there is no strategy leveraging merely a finite amount of past data at each step that guarantees getting any guesses right. In other words, for every strategy that leverages a finite amount of past data, there is a sequence of coin flips such that that sequence would result in the guesser getting every guess wrong. The proof uses the Compactness Theorem for First-Order Logic.