- Assume there are a finite number of possible universes . Let be a reference class of finitely many agents in those universes, and assume you are in . Let be the reference class of agents subjectively indistinguishable from you. Then SIA using is independent of as long as .
Let be a set of universes for some indexing set , and a probability distribution over them. For a universe , let be the number of agents in the reference class in .
Then if is the probability distribution from SIA using :
We now wish to update on our own subjective experience . Since there are agents in our reference class, and have subjectively indistinguishable experiences, this updates the probabilities by weights , which is just . After normalising, this is:
Thus this expression is independent of .
Given some measure theory (and measure theoretic restrictions on to make sure expressions like converge), the result extends to infinite classes of universes, with in the proof instead of .