I think a related point is touched on in my paper "Complexity and
Emergence", as in "Why Occam's razor". In both of these cases, one is
selecting from an infinite discrete set (of descriptions), which have
a uniform measure associated with them.

The answer you get, is the the probability of selection is related in
a simple way to the complexity of the description, as perceived by the
observer. There are no paradoxes, but the results are observer
dependent in a way that makes a lot of scientists uncomfortable.


Hal Finney wrote:
> 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?
> 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

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