OK, the “surreals” contain the transfinite ordinals, hence they contain the infinite cardinals as well. So, surreals can indeed model universes of strictly infinite size i.e. not just non-standard finite size.
I think the SIA problem of weighting towards the “largest possible” models still applies though. Suppose we have identified two models of an infinite universe; one says there are aleph0 galaxies; the other says there are aleph1 galaxies. Under SIA, the aleph1 model gets all the probability weight (or decision weight).
If we have a range of models with infinities of different cardinalities, and no largest cardinal (as in Zermelo Fraenkel set theory) then the SIA probability function becomes wild, and in a certain sense vanishes completely. (Given any cardinal X, models of size X or smaller have zero probability.)
Yes, this doesn’t solve the problem of divergence of expected utility, it just lets us say that our infinite expected utilities are not converging rather than only having arbitrarily large real utilities fail to converge.
OK, the “surreals” contain the transfinite ordinals, hence they contain the infinite cardinals as well. So, surreals can indeed model universes of strictly infinite size i.e. not just non-standard finite size.
I think the SIA problem of weighting towards the “largest possible” models still applies though. Suppose we have identified two models of an infinite universe; one says there are aleph0 galaxies; the other says there are aleph1 galaxies. Under SIA, the aleph1 model gets all the probability weight (or decision weight).
If we have a range of models with infinities of different cardinalities, and no largest cardinal (as in Zermelo Fraenkel set theory) then the SIA probability function becomes wild, and in a certain sense vanishes completely. (Given any cardinal X, models of size X or smaller have zero probability.)
Yes, this doesn’t solve the problem of divergence of expected utility, it just lets us say that our infinite expected utilities are not converging rather than only having arbitrarily large real utilities fail to converge.