> But if you talk about proportions or the probability of choosing an element from a given subset (what I suppose you actually mean by frequency), then this is exactly the way you define these things in Mathematics when dealing with infinite sets.

The phrase “from a given subset” is catching my attention. Are you talking about defining a probability measure on a finite subset of an infinite set? Because if so, that of course wouldn’t bear on the core issue being discussed of whether and how a unique probability distribution could be defined over an entire infinite set - but I am probably missing what you’re truly aiming at so maybe you can clarify.

You’re talking about cardinalities of infinite sets, which is not directly relevant to defining proportions or frequencies of the elements within an infinite set.

How does this help define the frequency of an element within the infinite set?

Not a single thing in that comment “corrected” what I said previously. I made a point only about infinite sets without natural orderings. I didn’t even argue whether an ordering can be given for an infinite multiverse. I noted that their response is interesting and potentially valuable for providing such natural orderings on infinite multiverses.

The point I made stands: if we cant find natural orderings for infinite multiverses, then we can’t meaningfully talk about the frequencies or proportions of universes within the multiverse. Their comment is germane to the antecedent (“if we can’t”). If they’re right, then we can indeed find natural orderings for infinite multiverses, so the consequent doesn’t necessarily apply.

The examples you give are interesting ways of recovering a natural ordering. It makes me wonder, in the case of spatiotemporally disconnected cosmological multiverses, if some kind of n-dimensional “similarity measure” could be used in principle, with our universe as the reference.

Of note, though, this…

> (Just like it is obvious that infinite coin flips should be time-ordered when referring to their “frequency”.)

… is problematic unless there is some actual infinite sequence of coin flips that we can refer to. Any hypothetical infinite sequence of coin flips could have any hypothetical time-ordering, so the original problem just rearises in the form of specifying the order of the flips in time.

> You can have “most” of infinity. For example, most whole numbers are not prime, and both of those sets are (the same size of) infinite. Still, the ratio of the number of primes to the number of non-primes below a certain value is small (and tends to zero as that value tends to infinity).

The difference between asymptotic density vs proportions of infinity is relevant here. Most numbers are non-primes only in the first sense, not the latter. Problem is, the asymptotic density depends on the ordering of the set, and not all infinite sets have natural orderings like the number line does.

For example, you can’t simply take the asymptotic density of an infinite set of coin flips to get frequencies of 0.5 for both heads and tails, because that depends on the ordering of the set being {H,T,H,T,H,T…} or {H,H,T,T,H,H,T,T,…}, or similar. But there’s no reason to privilege that ordering over {H,H,T,H,H,T,H,H,T,…}, which will give an asymptotic density of 0.67 for heads and 0.33 for tails. It might seem like something’s wrong with that last ordering, like we’d eventually run out of H’s or something, but in an infinite set we won’t ever run out.

The lesson is just that you can’t define frequencies or proportions in infinite sets that lack natural orderings. The number line is the exception, not the rule.

> First of all a theory is the hypothesis with the most proof. It doesn’t mean it’s proven.

This is a bit of a bugbear of mine. A theory is not what a hypothesis graduates into when it collects enough evidence. Theories are broad explanatory frameworks. They incorporate and generate many hypotheses/predictions. Some are highly backed up by evidence (Darwinian evolutionary theory, general relativity, etc.) and some are discredited (lamarckian evolutionary theory, etc.), and still others are currently speculative and not yet decisively confirmed nor disconfirmed.

Advaita Vedanta, even.