Given the irrationals are an uncountable infinity, (on, say, (0,1),) the intervals between the irrationals which partition the interval (0,1) must also be an uncountable infinity. However, on each interval between any two irrationals, we know that there lies at least a countable infinity of rationals.

Since there are an uncountable number of intervals, each with at least a countable infinity of rationals on them, how can there only be only a countable infinity of rationals?