[thaumasiotes]

Corollary: rational numbers must serve double duty separating many different pairs of irrational numbers.

You can keep going with things that sound weird:

There are infinitely many rationals between any two irrational numbers.

There are as many rational numbers between any two finite irrational numbers as there are between positive and negative infinity.

[throwawaymath]

Can you clarify that last point? I don't think it's strictly true. Aside from countable and uncountable infinities, you can have larger and smaller infinites as well. Unless every set S of all rational numbers between any irrational x and irrational y is isomorphic to the set P of all rational numbers, I don't see that this is correct. And I don't immediately see that you can put them into 1-1 correspondence.

[zeeboo]

Do you agree both sets are countable? If so, any two countable sets can be put into bijection by composing their bijections to the naturals.

[throwawaymath]

Oh, right. Yeah I guess that makes sense by definition.

[aportnoy]

Any infinite set is either countable or uncountable.

[throwawaymath]

I'm aware, but that's not what I meant.