If we are using Solomonoff induction, won't the expected population be infinite?... There might be a better story with a less crude treatment, but frankly I'm doubtful.

Looking back at this, I've noticed there is a really simple proof that the expected population size is infinite under Solomonoff induction. Consider the "St Petersburg" hypothesis:

Sh == With probability 2^-n, the population size is 2^n .... for n = 1, 2, 3 etc.

This Sh is a well-defined, computable hypothesis, so under the Solomonoff prior it receives a non zero prior probability p > 0. This means that, under the Solomonoff prior we have:

E[Population Size] = p.E[Population Size| Sh] + (1-p).E[Population Size| ~Sh]

Assuming the second term is >= 0 (for example, that no prior hypothesis gives a negative population size), this means that E[Population Size] >= p.E[Population Size| Sh].

But E[Population Size| Sh] is infinite, so under the Solomonoff prior, E[Population Size] is also infinite.

This shows that SIA is incompatible with Solomonoff induction, as it stands. The only way to achieve compatibility is to use an approximation to Solomonoff induction which rules out hypotheses like Sh e.g. by imposing a hard upper bound on population size. But what is the rational justification for that?