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AlephNeil comments on Shock Level 5: Big Worlds and Modal Realism - Less Wrong
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Sure.
So we agree: far from there being 'only finitely many copies', if there's any sense at all to be made of the 'number of copies' then it is infinite and any (or 'nearly any') possible configuration of the matter making up your body gets some non-zero density or probability depending on whether by configuration we mean a 'point' in phase space (if that's the right term...) or a little 'cube'.
Your original post is correct. I think "continuum-many" is misleading, but I can't object much given the context. I don't remember what I was thinking.
ETA: I would think of an eigenbasis and say that there are countably many electrons, none of which is localized.