>>14018694Well if you construct the rational numbers as equivalence classes of pairs of integers with second one nonzero (a,b)~(a',b') iff ab'=a'b then yes, the rationals are countably infinite, but if instead you use canonical forms where a rational number is a pair of integers (a,b) where b is positive and a,b share no common factors then rational numbers are finite (in fact they each consist of a single element).
Similarly you could easily come up with constructions in which they are uncountably infinite
