I'm reading "Understanding Analysis" by Abbott, and I'm confused about the density of $Q$ in $R$ and how that ties to the cardinality of rational vs irrational numbers.
First, on page 20, Theorem 1.4.3 "Density of $Q$ in $R$" Abbot states:
For every two real numbers a and b with a < b, there exists a rational number r satisfying a < r < b.
For which he provides a proof.
Later, on page 22, in the section titled "Countable and Uncountable Sets" he states:
Mentally, there is a temptation to think of $Q$ and $I$ as being intricately mixed together in equal proportions, but this turns out not to be the case...the irrational numbers far outnumber the rational numbers in making up the real line.
My question is: how are these two statements not in direct contradiction? Given any closed interval of irrational numbers of cardinality $X$, $A$, shouldn't be the case that we would have corresponding set of $X-1$ rational numbers, $B$, where each rational in $B$ falls "between" two other irrationals in $A$?
If this is not the case, how do we have so many more irrationals than rationals while still satisfying our theorem that between every two reals there is a rational number?
I know there are other questions similar to this, but I haven't found an answer that explains this very well, and none that address this (perceived) contradiction.