> When we make statements such as the size of the set of all natural numbers 1, 2, 3... is the same as the size of the set of all natural even numbers 2, 4, 6..., despite the former containing the latter but not vice-versa... it seems the word "size" -- and associated terminology "larger than", "smaller than", etc. -- is a particularly unhelpful set of words to have chosen for this.
It seems to me, when you're counting things, you wouldn't care what are the things you're counting specifically; while in your example it does matter for determining the subset relation. Whatever way of counting where it matters would be kind of weird.
> it seems like an unwarranted leap to go from this formal comparison of cardinality of infinite sets, to the intuitive English-sentence idea that "almost all" real numbers are irrational
But the article uses "almost all" in the formal sense? Which, by the way, also has pretty intuitive meaning, in my opinion.
> But the article uses "almost all" in the formal sense? Which, by the way, also has pretty intuitive meaning, in my opinion.
By formal sense, do you mean everywhere but a finite-measure set? Zero-measure?
I'll use finite-measure because it allows for intuitive constructions like "almost all real numbers are outside the closed unit interval 0 <= x <= 1".
But, there are still some constructions that a layperson might expect to hold, like: "almost all real numbers have fractional part < 1e-100", or "almost all positive numbers are of the form x.y with 0.y < 1/x" (thanks, harmonic series).
I think that without formal training, we're especially bad at reasoning about dense sets such as the set of rational numbers, compared to, say, the reals.
> ... intuitive constructions like "almost all real numbers are outside the closed unit interval 0 <= x <= 1".
But that would be wrong wouldn't it? I can produce all real numbers by pairing a finite number of (in this case 3) real numbers outside the set with each real number inside the set.
For each real x, in 0 <= x <= 1, we also have:
1/x (covers all real x, 1 <= x <= +infinity
-x (covers all real x, -1 <= x <= 0
-1/x (covers all real x, -infinity <= x <= -1)
The cardinality of all those reals outside of 0 <= x <= 1 is therefore 3x the cardinality of those inside 0 <= x <= 1, in this construction. But for infinite cardinalities the 3 can be discarded.
So there are exactly as many real numbers in 0 <= x <= -1 as outside it.
I don't disagree that these sets are the same cardinality. But cardinality isn't the only way to describe the "size" of a set.
I suppose the typical measure-theoretic definition of "almost all" / "almost everywhere" insists on "everywhere but a zero-measure set", and you can't define a measure that satisfies sigma-additivity that treats intervals of finite Lebesgue measure as such, while ascribing nonzero measure to sets of infinite measure.
But even so, the Lebesgue measure of R is infinite, while the same measure of the unit interval is 1.
If I do it the "normal" way (repeating the [0, 1] interval infinitely many times), there are infinite times as many reals outside the [0, 1] interval as there are in it.
But that "infinite times" is a countable infinity - the number of integers. How does "the number of reals in [0, 1] times the number of integers" compare to "the number of reals in [0, 1]"? Are they the "same" infinity?
What if we use rationals instead of reals? We can do the same x 3 thing, right? But the number of rationals is countably infinite, and "3 times countably infinite" is the same as "countably infinite times countably infinite", isn't it?
Maybe not the most formal of meanings, but my favorite is a probabilistic one: given a random element, how likely it is that it satisfies a predicate? If some elements don't, but it's still satisfied with probability 1, that's pretty clearly almost always.
EDIT: yeah you guys are right, I wouldn't worry too much about the prior not being a proper distribution, but still - this doesn't seem related to the cardinality of sets in a simple way after all!
If you are referring to a measure then 'almost all' pretty much exclusively means every except for a zero-measure set. I've never encountered another definition when dealing with measures.
But consider a different, entirely valid context of "almost all":
"Almost all natural numbers are greater than 10."
"Almost all prime numbers are odd."
If we wanted to extend this intuitively, we might want to support the statement that "almost all positive reals are greater than 10". One option of doing that is by using the nonstandard definition of "everywhere but on a finite-measure set".
Is the meaning unintuitive, or is it in fact simply that the world doesn't match your intuitions?
'cos I mean, I'm not a betting man, but "I bet mathematical facts are correct despite being unintuitive" makes my (winning, profitable) bet in December 2020 that Trump lost look like a true gamble by comparison.
It seems to me, when you're counting things, you wouldn't care what are the things you're counting specifically; while in your example it does matter for determining the subset relation. Whatever way of counting where it matters would be kind of weird.
> it seems like an unwarranted leap to go from this formal comparison of cardinality of infinite sets, to the intuitive English-sentence idea that "almost all" real numbers are irrational
But the article uses "almost all" in the formal sense? Which, by the way, also has pretty intuitive meaning, in my opinion.