In the following I shall use W0, W1, ... to denote different possible worlds; with "possible" I mean "compatible with physics as we know it".
In other words, each Wi corresponds to a particular 4-geometry and matter content.
Let me assume that "physics as we know it" does allow the creation of baby universes; see also this.
In the following I will use [Wb] to denote a world which created Wb as a baby universe; obviously many different worlds would be able to create the same world Wb and I leave it up to you if [] picks a particular one or represents all of them (it does not really matter for the argument I am trying to make).
We can then generalize this notation so that [Wa, Wb, ...] denotes a world which creates several baby universes Wa, Wb, ... and
[[Wc]] denotes a world which creates a baby universe which then creates Wc as its baby universe. And so on and so forth.
If you read my previous blog post, then you already know where I am going with this:
Let us assume that we can count all possible worlds in the set S = {W1, W2, W3, ...}.
It is then clear that every subset {W1, {W2, W3}} etc. corresponds to a possible world [W1, [W2, W3]] etc. etc.
so it immediately follows that S cannot be countable, because the powerset of S has higher cardinality than S itself.
It actually follows that S is not a proper set imho.
So how can a believer of the many worlds interpretation define a wavefunction of the universe over all possible worlds?
There is a different way to arrive at the same conclusion: If one considers a path integral Z over all 4-geometries (after some necessary but currently little understood regularization 8-) as the wavefunction of the universe, then the assumption of "baby universes" is equivalent to a sum over 'not simply connected' 4-manifolds; but this sum Z does not really exist, due to Goedel's theorem.
Notice that the problem does not go away even if we keep the 'not simply connected' 4-geometries (quasi)classical and only consider the quantum matter to trigger the creation of baby universes (or not).
4 comments:
But physicists have no problem to work with ill-defined math.
Very true, but I think every now and then one should point it out ...
I think the baby universes created by our world would be causally disconnected from us and therefore have little or no effect on the wavefunction of our universe.
As I see it, the reduction of the wavefunction to the one real world is pretty much the Copenhagen method.
But in general, the problem with too many worlds goes of course away if one assumes that baby universes are not really real (because they are causally disconnected), just as Wittgenstein's problem goes away if one assumes that facts about facts about facts... are not really facts.
But if one considers "the totality of all facts" or the reality of baby universes then there is a problem imho.
E.g. what is the probability that there is a doppelganger of me, i.e. an (almost) exact replica of the matter configuration sitting in front of a laptop typing this sentence, somewhere in the multitude of baby universes?
What is the probability that there is somewhere a machine, which simulates a dog, who dreams to be me? What is the ratio of worlds with such a machine to worlds with plain doppelgangers?
I am afraid there are too many worlds to calculate anything and to be honest this bothers me a bit...
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