[tashi]

If we want to open the floodgates on being too pedantic, I think there are uncountably more irrational numbers close to any rational number than there are rational numbers close to an irrational number. But in both cases, it's definitely a bunch.

[dllthomas]

> If we want to open the floodgates on being too pedantic

It's math. There's no such thing as "too pedantic", as long as you're being interesting and not mean about it.

> I think there are uncountably more irrational numbers close to any rational number than there are rational numbers close to an irrational number.

I think that's right.

Irrationals near a rational are almost certainly uncountable, as otherwise I think we can force all the irrationals to be countable by bucketing them. I think that concern is countered if any bucket has to be uncountable, but if it's not all that makes some rationals special in a way they probably aren't.

Rationals near an irrational is definitely countable, as all the rationals is countable.