Suppose that out of a set of infinitely many people, including you, a finite subset is chosen at random. How likely are you to be in that subset? Intuitively, not very likely. And the larger the infinity, the less likely.
But how do you pick out a finite subset at random? Here’s a natural way. First, pick out a subset at random, by flipping a fair coin for each person in the original set, and including a person in the subset if the comes up heads. Almost surely, this will generate an infinite subset (a consequence of the law of large numbers). But suppose this experiment is repeated—perhaps uncountably infinitely often—until the set picked out is finite. (This construction requires that the set of potential repetitions be well-ordered.) Or maybe you just get lucky, and to everybody’s surprise the set picked out is finite.
So now we have a method for picking out a finite subset at random (though it may take some luck). How likely are you to be in that finite subset?
Well, think about it step-by-step. Before you learned that the set picked out by the heads was finite, your probability that you were in the set was the probability that your coin landed heads, i.e., 1/2. Then you learn that the set of people for whom heads was rolled is finite. But this fact tells you nothing about your coin toss. For the claim that the set of people with heads is finite is logically equivalent to the claim that the set of people other than you with heads is finite. And the latter claim tells you nothing about your coin toss.
So, your probability needs to stay at 1/2.
Thus, the probability that a random finite subset of the infinitely many people includes you is finite. This is a little counterintuitive when the infinity is countable. And it becomes far more counterintuitive the larger this infinity gets. It is a stupendously implausible claim when that infinity is large, say ℶω.
Causal finitism blocks the story by making it impossible for you to find out that the set of people who got heads is finite.