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PussyStapler t1_jc5o5ry 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).

You chose an interesting example, because the ratio gets smaller as the limit gets larger, but when the limit is infinity, then the ratio becomes 1:1. The number of primes is the same as the number of non-primes. They correlate on a one-to-one correspondence (bijection).

Similar to the sum of 1+2+3+4+5+6+7..... gets bigger as n increases, but when n is infinity, the sum is -1/12.

Infinity breaks a lot of our expectations.

You may still have a point about it still is possible to have the "most of infinity," whatever that means, but your example doesn't work out. Just because something holds at really big values of n doesn't mean it holds for infinity.

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Ok_Tip5082 t1_jc5t5xg wrote

That said there are still different levels of infinity. The reals have a Lebesgue measure strictly greater than the rationals.

Also the "sum of 1..inf == -1/12" is not the case at all, the whole point of that example is to show how different contexts and definitions can have conflicting answers, similar to 0^0 or 1^(inf)

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lwalker043 t1_jc61hvm wrote

i agree with your decision to bring lebesgue measure into the conversation of "most", but i dont think that's the best example since the rationals have lebesgue measure as well as cardinality less than the reals.

a better example may be the cantor set and the reals: they have the same cardinality and yet the cantor set is measure zero where of course the reals have infinite measure. i think it's simple and fair to say that if "life" universes make up something like the cantor set to the reals, you have a very solid interpretation of "most" universes not having life at all.

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Ok_Tip5082 t1_jc6amn6 wrote

Yeah, you bring up some great points. Honestly I would want to go the opposite direction though and compare growth rates of functions, many classes of which tend to infinity but at vastly different rates.

I totally tried to get a better example but then went on a wiki binge and got lost around the page of hyperbolic growth which contrasts itself against exponential and logistic growth then found my way to robert miles again,....

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ImOpAfLmao t1_jc7euoy wrote

Firstly you're wrong when you comment that bijection implies ratio is 1:1. A quick example is the two sets A = {0, 1, 2, 3...} and B = {0, 2, 4, 6..}. These sets form a bijection (a -> 2a, and b -> (1/2)b). The ratio of #s in B less than x over #s in A less than x for some x is (ceiling(x/2) / x). And so lim x-> inf of (ceiling(x/2) / x) = 1/2. So bijections don't imply ratio is 1:1.

Secondly, you're wrong about their example not working out. The ratio in their example does not become 1:1, the limit tends to 0. Quick explanation, simpler ways exist, but just for illustration:

From the prime number theorem, we know lim x-> inf of (pi(x)/(x / (log x)) = 1, where pi(x) counts the number of primes below x.

In this case we want to figure out what limit of the ratio of primes is to non primes below x, or lim x->inf of (pi(x)/(x - pi(x)). Dividing both numerator and denominator by (x/(log(x)), we have lim x-> inf of pi(x)/(x/(log(x)) / ((x - pi(x))/ (x / log(x)).

Quotient limit rule, so the numerator limit is 1 by prime number theorem, so we have it equivalent to 1 / (lim x-> inf of ((x - pi(x)) / (x / log(x))). So if we show the denominator limit goes to infinity, the entire limit is 0.

Bottom limit simplifies to lim x-> inf of x*log(x)/x - log(x) * pi(x)/x, the latter term goes to 1 by the prime number theorem, and thus the entire denominator limit is just lim x-> inf of (log(x) - 1) which goes to infinity, thus entire limit goes to 0.

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