You've existed for an infinite amount of time, and each day until the year 2100 a coin is tossed. You always know the results of the past tosses, and before each toss you are asked to guess the next toss. Given the Axiom of Choice, there is a mathematical strategy that guarantees you make only a finite number of mistakes.
Here's a simpler fact, no doubt well-known, but not dependent on the Axiom of Choice. There is a mathematical strategy that guarantees that you guess correctly infinitely often. This is surprising. Granted, it's not surprising that you guess correctly infinitely often--that is what you would expect. But what is surprising is that there is a guarantee of it! Here's the simple strategy:
- If among the past tosses, there were infinitely many heads, guess "heads"
- Otherwise, guess "tails".
I take the paradoxical existence of this mathematical strategy to be evidence for causal finitism: causal finitism rules out the possibility of your having observational information from infinitely many past tosses. Thus the strategy remains purely mathematical: it cannot be implemented in practice.