Hacker News Reader

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.

View on HN

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.

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.

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

Any infinite set is either countable or uncountable.

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

Reply on news.ycombinator.com