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Anthropics in infinite universes

1Charlie Steiner

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A similarly odd question is how this plays with Solomonoff induction. Is a universe with infinite stuff in it of zero prior probability, because it requires infinite bits to specify where the stuff is? Quantum mechanics would say no: we can just specify a simple quantum state of the early universe, and then we're within one branch of that wavefunction. And the (quantum) information required to locate us within that wavefunction is only related to the information we actually see, i.e. finite.

When talking about anthropics, people often say things like "assume the universe is finite; weird things happen in infinite universes". I've myself argued that SSA breaks down when we encounter infinities; SIA breaks down sooner, when we encounter expected infinities.

You can formalise this informally

^{[1]}with the thought that:A superficially convincing argument; but not one you'd use for anything else. For instance, consider the following:

I've argued before that anthropic questions are pretty normal. Why would we accept the reasoning in question 1, but reject it in question 2?

We shouldn't. We can deal with questions like 2 by talking about limits of probabilities in larger and larger spaces, or by discounting distant observations (similar to sections 2.3 and 3.1 in infinite ethica). So we might define conditional probabilities like P(X∣Y) in an infinite universe in the following way:

Note that this definition works just as well for Y= "we observe the force of gravity to be blah" as with Y= "we exist".

Now, that definition might not be ideal (in particular, "radius" is not defined for relativistic space-time). No problem: different definitions of probability are asking different questions, and can lead to different anthropic probabilities, just as in the finite case.

I'll call these class of questions "SIA-limit questions", since they are phrased as ratios of observers, and dependent on how we use limits to define probability in infinite universes. They each lead to various "SIA-limit anthropic probability theories"; in most standard situations, these should reach the same answers as each other.

Yes, it's perfectly possible to formalise informally, and I encourage people to do it more often. ↩︎