Consider a class C of possible universes where, at t0, exactly 100 independent random processes are activated, and nothing else random happens. Suppose there are no qualitative differences between universes in C other than those due to the differences in the outcomes of the processes. Suppose, further, that each process can result in either a "heads" or a "tails", with equal probability 1/2.
There are, thus, exactly 2100 different types of possible universes in C (where we say that universes are of different type provided that they're not exactly alike).
Suppose we live in a multiverse that contains exactly n universes from C. If n<2100, then there are some possible types of universes not represented in our multiverse—there are some combinations of heads and tails in a C-type universe that don't occur in our multiverse. If n>2100, then there are some types of C-universes that are multiply represented in our multiverse.
So if every type of universe is represented exactly once, there are exactly 2100 C-universes in the multiverse, and each has a different heads-tails profile. But how likely is it that each would have a different heads-tails profile? Assuming that what happens in different universes is stochastically independent, this is just a version of the birthday problem. If n=2100, then the probability that each of the n universes has a different one of the n profiles is n!/nn, which according to Stirling's formula is something like exp(−2100). That's a very, very tiny number. And since classes like C can be found for any number of random processes, not just 100, it follows that the probability that every type of universe is exemplified exactly once in the multiverse is smaller than every positive real number, i.e., it's zero or infinitesimal.
So if we live in a multiverse, almost surely either some possible types of universes aren't instantiated or some are multiply instantiated or both.
8 comments:
I was going to correct you about that very small number that looked suspiciously close to one, but you fixed it. :)
Hey, 1 is small. :-)
And that's why I don't think "multiverses" provide any challenge to fine-tuning arguments. You can't expect to get Hamlet out of a computer program that produces random sequences of letters; but you could if it were programmed to produce every possible sequence. Similarly, if there's no pattern to the multiverse, we can't rely on its producing a universe like this one. And if there is a pattern — if, say, it is designed to produce all (or most) possible universes — then it is itself "tuned" in some way.
Alex:
Can you find me one of those 2**100 universes with the following characteristics - all full of Boone and Crockett deer (deer season was a bummer) a sky full of geese (goose season was too warm and they didn't come down); no need to work (work interferes with hunting), and no sinus infections or head colds (I've lost most of this weekend to one). I know I'm asking for a perfect world but with 2**100 universes the chances of finding one are good. :-)
I think there is something wrong here. You are assuming that the multiverse theory is a probalistic one, but it isn't.
There are many multiverse theories. The ones that have a chance of being true had better be probabilistic.
I think this is related to the idea of modal realism isn't it? After all, a supporter of modal realism makes the claim that there is no reason a single world should be privileged and makes the logical leap to conclude that all possible worlds are actual. But this post made me realize that by their logic, there is no reason for any world to be actualized only once, so a modal realist would have to explain why we shouldn't expect an infinite amount of worlds that are exactly the same as each other.
Lewis explicitly leaves it unanswered whether there are duplicate worlds.
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