Something I've thought about for a bit on an intuitive level is about the countability/density of Q. It seems almost contradictory. I know what countable means, I know what dense means. I know how to prove Q is dense. I know how to prove Q is countable. But on an intuitive level, it's hard to wrap my mind around it. Intuitively, if something is countable, given any element in the set and enough time, you can count until you reach it. Like for N, given ANY natural number, no matter how big, and infinite time, and I could count until I reach it. But with Q, lets say I wanted to count to 1. There's an infinite number of elements between 0 and 1, so I would be there forever, never being able to reach 1? What about 1/2? I shouldn't be able to reach there. The way to reconcile it in my head is that the bijection with N isn't order preserving, right? In the way you count Q on [0,1], you would superficially go through it, then again, and again and again. It's still a little unclear though. Anyone able to reconcile them intuitively?