Start with these two countably infinite multiverses:
- Before any universe of multiverse 1 exhibits life, a fair die is cast in each universe. If the die shows six, a diamond core forms on an uninhabited moon of one earthlike planet. Otherwise, an iron core forms there. Then, completely independently of the die roll and the core of the moon, a single person comes into existence on that earthlike planet in each universe of that multiverse. The person is then informed of all of the above facts, as well as of the fact that infinitely many dice showed sixes and infinitely many did not (that fact was very likely, but now the person is sure).
- Multiverse 2 is just like multiverse 1, except that when the die shows six, an iron core forms; otherwise, a diamond core forms.
Question: What probability should the persons in the multiverses assign to the proposition that the moon of their planet has a diamond core?
Intuitively, in multiverse 1, the probability should be 1/6, and in multiverse 2, it should be 5/6. But now notice this. At the time that the person is asked to assign the probability, the two multiverses are exactly alike in all relevant features, and the persons know that. In each multiverse, there are infinitely many earthlike planets containing a person and having an uninhabited moon with a diamond core, as well as infinitely many where the moon has an iron core. The history of the two multiverses is different, but why should the history matter when the outcomes are relevantly the same? This line of thought suggests that the persons in the two multiverses should assign the same probability to their moon having a diamond core. Presumably that same probability will be neither 1/6 nor 5/6, but may be 1/2 or an interval or just plain undefined. This is a counterintuitive conclusion, but it is hard to avoid.
Now let's consider two more multiverses:
- The same as multiverse 1, except that the dice are cast and the moon cores formed after the persons come into existence.
- The same as multiverse 2, except that the dice are cast and the moon cores formed after the persons come into existence.
So, by the counterintuitive conclusion about multiverses 1 and 2, the people in multiverses 3 and 4 should make the same probability assignments about their moons' compositions. In other words, their probabilistic reasoning is undercut.
If this is right, however, then if it turns out that we live in an infinite multiverse, our normal probabilistic reasoning is undercut. In particular, any science based on probabilistic reasoning--and all modern science is like that--that concludes to an infinite multiverse has undercut itself.
Objection: Being informed that there are infinitely many iron cores and infinitely many diamond cores in multiverses 1 and 2 violates causal finitism: the informant's words need to depend on infinitely many events.
Response: It shouldn't matter whether one is or is not informed of the fact, since the fact has probability 1, and learning a probability 1 fact shouldn't change one's probabilities.