The axiom of choice was just to justify how it's possible to be born now even if there are infinite people, which is an objection I have seen before, though you didn't make it. Also it's a fun reference to make.

It's possible I'm wrong, but I'll try to walk you through my logic as to why it isn't informative that we're born now. Imagine the "real" set of people who ever get born. When choosing from this bunch o' people, the variance is <N^2> - <N>^2 = N^2 /3-N^2 /4 = N^2 /12 (for large N).

So the standard deviation is proportional not to the square root of N, as you'd expect if you're used to normal distributions, but is instead proportional to N. This means that no matter how big N is, the beginning is always the same number of standard deviations from the mean. Therefore, to a decent approximation, it's not more surprising to be born now if there are 10^20 total people born than if there are 10^100.

Of course, writing this out highlights a problem with my logic (drat!) which is that standard deviations aren't the most informative things when your distribution doesn't look anything like a normal distribution - I really should have thought in two-tailed integrals or something. But then, once you start integrating things, you really do need a prior distribution.

So dang, I guess my argument was at least partly an artifact of applying standard distributions incorrectly.