I understand that there are infinitely many more irrational numbers than there are rationals - there are many ways in which one can intuitively understand this. However, consider the following:
LEMMA: For every pair of distinct irrational numbers, there exists a rational number between them.
PROOF: Let $a,b$ be distinct irrational numbers such that $b>a$ and let $\delta = b-a$. Then for some large enough $N$, we have $N\delta >1$ so there exists an integer in the interval $[Na,Nb]$ and as such there exists a rational number in the interval $[a,b]$.
By this logic, we can find a rational number for every pair of distinct irrational numbers - why does this not imply there are as many rationals as irrationals?