David Deutsch wrote about the Self Selection Assumption, on the
Fabric-of-Reality list:
> One problem with both of these is that there is no preferred meaning to
> sampling *randomly* from an infinite set, except in certain very
> special cases.
>
> A discrete infinity of copies of me is not one of those cases, so I
> don't think it is meaningful to select randomly from the "set of all
> observers who will ever be created who are (in any sense) like me". So
> doesn't the thing fall down at the first hurdle?

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First, I'm not so sure it is true that you can't select randomly from an
infinite set. In the level 1 multiverse, there are an infinite number of
copies of me. On some philosophical perspectives, I am exactly one of
those copies. This represents an actual choice from an infinite set.
Anyone who accepts both of these principles (the level 1 multiverse
and the fact that he is actually in a single location) must accept the
possibility of random selection from an infinite set.
But suppose we do accept that this is impossible. The SSA could still
work if it turned out that there were only a finite number of observers
and observer-moments in the reference class.
This is plausible if two conditions hold. The first is that each
observer is described by only a finite amount of information. This is
established by our current theories of physics. The second is that
there is an upper limit on the size an observer could have; that is,
that there are no infinitely or arbitrarily large observers. That is
a little harder to defend, but for an observer of arbitrary size to
come about, the universe (its universe, that is) would have to last
subjectively forever, and have an infinite amount of information in it.
Therefore I think it is rather problematic to suppose that there is no
upper limit on the size of an observer, as it requires infinities to
creep into the physics of the universe in several places.
If observers do have an upper limit, then there are only a finite number
of possible observers, and possible observer-moments, and Deutsch's
objection fails on that basis.
Hal Finney