There are infinitely many rational numbers between any two integers (or any two rationals), yet the rationals are still countable, so this reasoning doesn’t hold.
The only simple intuition for the uncountability of the reals I know of is Cantor’s diagonal argument.
anton@lemmy.blahaj.zone 3 weeks ago
There are also an infinite number of rationale between two integers, but the rationals are still countable and therefore have the same cardinality as the naturals and integers.