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:

1. In an infinite universe, anything can happen, no matter how unlikely: life must exist somewhere. So our existence doesn't tell us anything about life; its probability could be anything at all.

A superficially convincing argument; but not one you'd use for anything else. For instance, consider the following:

2. In an infinite universe, anything can happen, no matter how unlikely: if gravity didn't exist, somewhere it must seem to exist by shear chance. So our observation of gravity doesn't tell us anything about gravity; its probability could be anything at all.

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 $P(X\mid Y)$ in an infinite universe in the following way:

• Let $P_{rl}(X\mid Y)$ be the ratio of observers, within a large hypersphere of radius $r$ centered on location $l$, that observe $X$ and $Y$, relative to the proportion that observes $Y$. If this tends to a limit as $r\to \infty$, independently of $l$, then define that limit to be $P(X\mid Y)$.

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.