I’ll just explain why I have come to this question here. I was thinking about the multiverse thing. It occurred to me that if the multiverse is real, as in an infinite quantity of universes, then there is an infinite number of universes where you exist in every variation and an infinite number of universes where you don’t exist in every variation. So if, at random, a portal links two universes together, there is a chance that you will link to a universe with another version of you. It seems like the probability of this would he low, but not zero, despite the fact there would be infinitely more universes without a version of you than there would be with one.
In: Mathematics
It’s zero.
If you have an infinite set and want each element to have the same probability to happen, then:
If the set is countable (i.e. you can count all it’s elements: first, second, thirst, etc), then it’s not possible since any reasonable definition of what it means to sum countable amount of summands will get you “countable sum 0+0+0+… is 0” and “if a>0, then countable sum a+a+a+… is infinite”. Since we want the sum of all probabilities to be 1 (or 100%), that won’t do. So if you want to assign probabilities to a countable set, they can’t all be the same (see two envelopes paradox).
If the set has continuum elements (e.g. dots on a line segment), then you can work with assigning 0 to all elements of this set– if you also add some way of assigning probabilities to at least some of the infinite subsets of this set. The most natural example:
You have a segment of length 1, you pick a dot, and we want all dots be in some sense equally probable. Then you say that probability is 0 for every individual dot, but you also say that probability to pick a dot from a given (sub)segment of length a is a. E.g. the probability that the dot is from the first half of the segment is 1/2 since that half has length 1/2.
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