That doesn't explain why rationals and reals are different. Rationals are a countable subset of the reals, to prove that you need to produce a method of enumerating them, not just say "they're a subset so they're smaller."

Lots of subsets of the reals are uncountable, most notably if the rationals are countable and you split the reals into rationals and irrationals, the irrationals have to be uncountable. Otherwise you'd be forced to say that the union of two countable sets could be uncountable, which isn't true (if you have two countable sets you can automatically produce an enumeration of the union).

I don't see how your point about countability invalidates what I said. Irrationals are uncountable, but there are fewer of them between 0 and 1 than reals between 0 and 1, by virtue of the fact that all irrationals are reals but not all reals are irrationals.

I admit I'm using "subset" in the colloquial sense, meaning "a set that's missing some of the elements in its superset", which is not technically the correct definition (every set is a subset of itself, i.e. a subset doesn't have to be missing anything). So technically a subset does not imply fewer elements. Is that what you're getting at?

I'm not particularly well versed in set theory, so maybe I'm just misunderstanding your point about countability.

You didn't say anything entirely incorrect, I think! But the important bit when it comes to talking about sizes of infinity is that, while both irrationals and rationals are smaller than the reals in one sense (they're both proper subsets), they're very different in another sense: you actually can index the rationals by the natural numbers whereas you can't do the same with the irrationals. That's the much more interesting sense in which the rationals are smaller (or, alternatively, that reals are bigger). The rationals are smaller than the irrationals in this sense as well, despite not being a subset of them at all.

I think I still don't understand 100%, but that's probably my lack of formal knowledge about this stuff. I appreciate the explanation.

Consider it from a measure theory point of view, at least colloquially. If you could assign a mass of 1 (lb/kg/g/whatever) to the reals and separate them into two piles: rational and irrational then weigh again. The irrational pile would weigh 1 and the rationals, 0. So yes you took something out, but also in some sense, it doesn't matter.

> Irrationals are uncountable, but there are fewer of them between 0 and 1 than reals between 0 and 1, by virtue of the fact that all irrationals are reals but not all reals are irrationals.

This is the important point where you are wrong. For a simpler to explain example : the naturals are a subset of the integers, and yet the set of all integers is exactly as big as the set of all naturals. We say this because we can actually find a natural number that corresponds to every integer number: 0:0, 1:+1, 2:-1, 3:+2, 4:-2,... A slightly more complicated construction exists for the naturals compared to the rationals: they are also the same size.

However, it can be shown that the same can't be done between the naturals and the reals, or even any real interval. So, by this definition, there are more real numbers between 0 and 1 than there are in the entire set of natural numbers.

Depends what you mean by 'indexed'. If you accept the axiom of choice then you can index the real numbers with some uncountable ordinal.

I meant it in the sense that I assume GP meant it, i.e., the index of an element in an ordered set is an integer corresponding to its position in that set, starting at 0 or 1 for the first element (depending on your indexing scheme of choice).

Does the axiom of choice let you repeatedly pick the same real numbers in the same order out of a set? Or do you need uncountably infinite storage to keep your list of indexes?

That's an interesting question, but kind of hard to answer. Programming is inherently constructive so approaching a non-constructive proof as a programming problem tends to be problematic.

I don't think you run into any problems if you let the axiom of choice pick the same numbers every time. It would be a bit problematic if you were to add that as an extra axiom (suddenly every proof let x,y in X then ... therefore x=y would become trivial).

> If you pick one and call it N, there's an infinite quantity of real numbers between 0 and N and between N and 1. Therefore it's impossible to assign an index to N.

This isn't "sloppy language", it's just wrong. Don't try to downplay spreading misinformation. It remains true that there are infinitely many rationals in (0, N) and (N, 1), and yet it is entirely possible to enumerate the rationals (i.e. assign indices).

> There's an infinite number of those between 0 and 1 as well, but because it's a subset, there are "fewer" of them.

Also misinformation. (0, 0.5) is a subset of (0, 1) yet the two sets have the same cardinality, whether we take them both in Q or both in R.

> This isn't "sloppy language", it's just wrong.

Yes, no less than four people have explained what's wrong with my comment. I made the "sloppy language" edit before I fully understood them.

> Don't try to downplay spreading misinformation.

> Also misinformation.

LMAO. Every incorrect thing you see on the internet isn't the work of a Russian troll or whatever you're trying to accuse me of here. Have you considered getting a life?