Georg Cantor also famously claimed that the cardinality of the set of rational numbers equals the cardinality of the set of integers because he could devise a 1-1 correspondence or mapping between them as shown here and here.
Relatedly, the continuum hypothesis is: There is no set whose cardinality is strictly between that of the integers and the real numbers (rationals and irrationals).
Accepting certain assumptions Paul Cohen showed that the continuum hypothesis is neither true nor false. I think it’s false, the quantity of rational numbers being greater than the integers, but less than the reals (rationals and irrationals combined). I base this on two perspectives different from that of Cantor and Cohen. The first is the real number line. There is an unlimited count of rational numbers between 0 and 1, between 1 and 2, and so forth.
The second is part-whole logic. As shown in the above links, a 1-1 mapping can be made. However, note the first column in the array in either link, which is the integers. A different 1-1 mapping or function f(x) = x can also be made. Accordingly, the integers comprise a proper subset of the set of all rational numbers, implying the set of rational numbers is larger than the set of integers.
“Cantor was concerned to combat the Aristotelian view that there cannot be an actual infinity, mainly because Cantor believed that God was infinite.” – Michael Huemer, Approaching Infinity, p. 71.