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In "Our Mathematical Universe" Tegmark claims that inflation theory implies existence of (countably, it seems to me) infinite set of universes. He says that from this follows existence of a universe in which a person almost identical to me has lived the same life and observed the same thing. Is this implication correct?

If we were talking about exactly the same universes, then it seems to me obtaining such a universe would be a probabilistic event of zero measure. And because there is only a countably infinite set of universes, we wouldn't get an exactly the same universe anywhere.

But I guess we can talk about universes differing from our universe by no more than epsilon. Interpreting this probabilistically (with a lot of handwaving, of course), is this definitely an event of nonzero measure? If yes, then is Tegmark correct that Inflation theory implies this?

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The argument is based on our observable universe having a finite, nonzero probability since there is only a finite number of distinguishable configurations within a finite radius (as given by the Bekenstein bound). This is your epsilon. So the measure of our local universe configuration is non-zero, and hence in a sufficiently large and randomly initialized universe (whether spatially infinite or an eternal inflation structure) there will be an infinite number of instances.

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  • $\begingroup$ Why is that ? Doesn't it follow that some universe will be repeated at least an infinite number of times ? Why are all possible universes repeated ? $\endgroup$
    – THC
    Commented Jul 8, 2019 at 12:24
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    $\begingroup$ The crucial part is the "randomly initialized" part - one could of course have one copy of observable universe 1, 2 of universe 2, 3 of #3 and so on. However, this configuration is extremely unlikely compared to the ones where the number of copies of a given observable universe tends to infinity as we look at larger and larger sets. This ergodicity assumption can be quite subtle: having a finite number of copies of some universes is possible but have measure zero, that is, probability 0 but not in the sense of being impossible, just never happening. $\endgroup$
    – Anders Sandberg
    Commented Jul 8, 2019 at 13:58
  • $\begingroup$ That is what I expected -- thank you for the clear explanation! I imagine that the behavior (with respect to substructures) of such a randomly initialized universe could be compared to the behavior (with respect to substructures) of a (countable) random graph ? $\endgroup$
    – THC
    Commented Jul 8, 2019 at 14:39

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