ℝ is mind boggling. Here's a puzzler at a layman+ (high school or freshman) level: Between two reals there's at least one rational number [1], and between two rationals there's at least one real number. So there must be an equal number of them. (You can replace "at least one" with "infinitely many" and it's still true.)
[1]: https://math.stackexchange.com/questions/2602418/proof-there...