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Eugine_Nier comments on Multiverse and complexity of [laws of] the observed universe - Less Wrong
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Comments (16)
The question is not which set is larger, which is in any case almost meaningless since both are infinite, but which set has larger probability measure.
I do think it's meaningful to talk about different sizes of infinity (for example, countable vs. uncountable), but probability measure is probably more relevant.
To expand on that point - what you are refer to there as "different sizes of infinity" are different cardinalities of sets. As you note, what sorts of infinities you have to use depends on what you are trying to measure; raw cardinalities are rarely the right notion of size, here we want to think in a measure-theoretic context. But it's worth noting that for measuring other things different systems of infinite numbers must be used; cardinalities and "infinities" should not be identified.
Yes. I initially implicitly assumed uniform distribution (i.e. used it as default without any information to prefer some different particular one).
But what meaningful probability measures might be in this case? (Besides a point described few comments below [http://lesswrong.com/r/discussion/lw/393/multiverse_and_complexity_of_laws_of_the_observed/33li?c=1])
In some cases, there's no way to define a uniform distribution (i.e. over the integers), so you've got to do something else.
Huh, can you define an improper uniform distribution over the integers like you can occasionally for the real line? Or does that always lead to an improper posterior?