The subset of the real number system with the most rational numbers is the set of rational numbers themselves, due to their countably infinite nature and their definition as ratios of integers with non-zero denominators.
The subset of the real number system that contains the most rational numbers is the set of rational numbers itself. Rational numbers are defined as numbers that can be expressed as a ratio of two integers, where the denominator is not zero. Every rational number is, by definition, part of the real numbers, but not every real number is rational because the real numbers also include irrational numbers.
Rational numbers are countably infinite, meaning there is a one-to-one correspondence between the set of rational numbers and the set of natural numbers. Therefore, the set of rational numbers is as large as it gets within the realm of countable sets. However, in terms of density, both the rational and irrational numbers are dense in the real numbers, which means that between any two real numbers there are infinitely many rational and also infinitely many irrational numbers. This illustrates that while the rational numbers are vast, their presence in the real number system is interspersed with the irrational numbers.