Submitted by Azmisov t3_11qjmwp in philosophy
Seek_Equilibrium t1_jc5t3ev wrote
Reply to comment by EmptyTotal in A philosophical dive into “Everything Everywhere All at Once” by Azmisov
> 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.
throwawayski2 t1_jc6azmd wrote
>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.
You may need to explain a bit more, because as a mathematician that just seems plain wrong:
You can define sets of any proportion on any bounded set of R^n (an infinite set with no natural ordering for n > 1). That's a very basic thing in Measure Theory. For example you can just generalize the Cantor set to any dimension to get an infinite set of points that has no volume.
Edit: just some minor correction of my part
Seek_Equilibrium t1_jc8t769 wrote
How does this help define the frequency of an element within the infinite set?
throwawayski2 t1_jcb56n3 wrote
I didn't mention frequency if you read again. Frequency - at least in the probabilistic sense - requires a observational component, that is reasonable to assume when discussing possible worlds. But that has nothing to dobwith infinity but with the fact that you can't observe possible worlds.
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.
Seek_Equilibrium t1_jcbheol wrote
> 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.
throwawayski2 t1_jccd2r6 wrote
No, it can also be defined on infinite subsets. That's why I mentioned Cantor sets, because these are measurable uncountable sets, such that choosing an element from it (given uniform choice from the bounded set on which it is defined) has probability 0 (which is different from our finite intuition, that it is impossible).
It is basically just a generalization of the concept of volume.
EmptyTotal t1_jc6kbrc wrote
>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.
In the context of multiverses, there are natural ways of ordering them. In MWI for example, you could consider universes that first differ from ours by a more recent branching event to be "closer" than ones that branched further back. Then whatever density you want can be defined in the set of universes that diverged later than time t, as t is taken to zero.
Frequency in a multiverse shouldn't really be any less intuitive than a frequency measurement in our single universe. If space is infinite, then it also contains infinite planets. But it is still obvious that most of space is empty.
(Just like it is obvious that infinite coin flips should be time-ordered when referring to their "frequency".)
Seek_Equilibrium t1_jc7eikj wrote
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.
python_hunter t1_jc8099y wrote
TL;DR "thank you for the correction" ;D
Seek_Equilibrium t1_jc81w1w wrote
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.
HortenseAndI t1_jc9kdc8 wrote
Y'all are getting worked up about comparing countable infinities. There are other ways to have 'most' of an infinity - e.g. there are more non-rational reals than rationals because the former is uncountable
Seek_Equilibrium t1_jc9kzge wrote
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.
HortenseAndI t1_jc9t1yj wrote
I mention it because the relevant passage in the original article is "you cannot have “most” of infinity. The only scenario where it somewhat makes sense is where a finite number of worlds evolved life, but an infinite number did not.", which is blatantly untrue given that you can compare infinite sets with different cardinalities. My point is there's no need to get hung up on the probability space of countably infinite sets to comfortably assert that that's nonsense, which is what was happening here
python_hunter t1_jc80o1e wrote
I'm not sure that defining a 'sequence' is as relevant here as one might think; while it's a strategy often used in proofs, I don't think that (in my layperson's understanding) this precludes there existing different 'orders' of 'infinity'
DarkSkyKnight t1_jc68m1d wrote
Could just limit ourselves to the space of measurable sets. Seems like the natural approach since we're dealing with notions like "most" (almost), "frequent" (probability) here. And it doesn't seem immediately clear why we would need unmeasurable sets for the multiverse.
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