Very large multiverses and trans-world depravity

Consider this standard bit of dialectic. One gives a Free Will Defense relying on the logical possibility of Trans-World Depravity:

1. TWD₁: In every feasible world, some significantly free creature sins at least once.

But the response is: “Yes, this shows that the existence of God is logically compatible with moral evil, but since TWD is exceedingly unlikely to be true, this does not help to counter the argument that it is exceedingly unlikely that God would create a world with no moral evil.” (Note: TWD₁ is logically weaker than Plantinga’s TWD but does the job just as well. Cf. this.)

Why think TWD₁ unlikely to be true? Well, confine our attention to strongly actualizable world cores (i.e., the aspects of the world that God would strongly actualize) containing involving a hundred independent significantly free choices with agential motives balanced between good and evil. For any such world core, the chance that all the choices would go right if the core were strongly actualized is something like (1/2)¹⁰⁰. So we would expect one in 2¹⁰⁰ such world cores to have the counterfactuals of freedom come out favorably, i.e., with all the choices being right. But there are infinitely many such world cores—each with a different collection of agents (to ensure independence between the counterfactuals holding of each core)—and so the probability that in some core the counterfactuals come out right should be extremely high (namely 1, given real-valued probabilities).

Now here is an interesting next step in that dialectic. Instead of working with the TWD, work with this:

2. TWD_ℵ1: In every feasible world containing uncountably infinitely many significantly free choices, at least one of these choices is wrong.

And then add the plausible intuition that it is likely that God would want to create a world with uncountably infinitely many significantly free choices (e.g., because he would probably want to create uncountably infinitely many significantly free people, perhaps in a multiverse).

The improbability response is much harder to make against TWD_ℵ1 than against TWD₁. Remember that the argument against TWD₁ worked by generating a sequence of worlds with independent conditionals of free will each of which had a probability of (1/2)¹⁰⁰ of being a counterexample to TWD₁. But we can’t do that with TWD_ℵ1. Given an uncountable infinity of significantly free choices, we would expect the probability that all these choices would be right if the world core were actualized is zero: it’s logically possible, but it’s even less likely than tossing a fair coin for every day of an infinite life and getting heads each time (for an infinite life would have countably many days). Granted, there is an uncountable infinity of world cores to try. But if the chance of each one being a counterexample to TWD_ℵ1 is zero, without some special argument we can’t assume there is a meaningful and high probability that at least one is a counterexample.

Technical note: I opted for worlds with uncountably infinitely many significantly free choices, because if the worlds had countably infinitely many significantly free choices, it might be possible to make the “it’s extremely unlikely” argument go. Imagine a countably infinite sequence of independent significantly free choices, where the nth choice has probability 1 − 2⁻ⁿ of going right. Then the probability that all the choices will be right is actually about 0.29. Using worlds like that, one could produce an argument that TWD_ℵ0 (i.e., what we get when we remove “uncountably” from (2)) is very unlikely to be true.