According to the Many Worlds Hypothesis (MWH) our universe is one among many—perhaps an infinite number—of spacetime universes (misleadingly called “worlds”). It is not clear whether the MWH should postulate a large but finite number of worlds, or an infinite number of worlds. Here are two reasons to think the MWH should postulate an infinite number as opposed to a large but finite number:
(i) Postulating an infinite number of worlds constitutes a simpler hypothesis than postulating a large but finite number. This is because the latter requires an additional explanandum—viz., why this particular number?—in the same way telling a friend to meet at 6:13 or 6:21 rather than, say, 6:00 or 6:30 does.
(ii) If the only reason for postulating a large but finite number of universes is just to meet a certain probabilistic threshold (i.e., however many universes are needed to render more probable fine-tuning), then it is extraordinarily ad hoc.
I have not looked deeply into the literature on the MWH, so I am unaware of any reasons for postulating a large but finite number of universes independent of that mentioned in (ii) (if any reader is aware of such reasons, I’d appreciate a note).
So suppose the MWH postulates an infinite number of universes. If the normalizability objection (see here for the original statement by the McGrews and Vestrup, or Here for my brief synopsis) to fine-tuning arguments (FTAs) is sound, then it is also a sound objection to the MWH. This is because the probability range postulated by the MWH would be infinite, and hence, not normalizable.
Suppose instead that the MHW postulates however many universes—from one to infinity—is needed to render more probable fine-tuning. This shows that there is no principled upper bound on the probability range postulated by the MWH. And this is precisely the problem the normalizability objection highlights. So it’s not clear that the MWH can avoid the normalizability objection even if it can get away with postulating a large but finite number of universes.
In other words, one cannot endorse both the normalizability objection and the MWH in response to FTAs. My own suspicion is that this casts serious doubt on the soundness of the normalizability objection.