The trick is that there are an infinite number of rational numbers between zero and one. When dealing with infinite sets, one way to count their members is to put them into one-to-one correspondence with some standard set, like the set of natural numbers or the set of real numbers. These two sets (i.e., the naturals and the reals) have different sizes: it turns out that the set of natural numbers cannot be put into one-to-one correspondence with the real numbers. No matter how one tries to do it, there will be a real number that has been left out. In this sense, there are “more” real numbers than natural numbers, even though both sets are infinite.
Thus, a useful classification for infinite sets is as “countable” (can be put into one-to-one correspondence with the naturals) or “uncountable” (too big to be put into one-to-one correspondence with the naturals). The rational numbers are countable, so any infinite subset of rationals is also countable. When CronoDAS says that there are as many rationals between zero and X as there are greater than X, he means that both such sets are countable.
The trick is that there are an infinite number of rational numbers between zero and one. When dealing with infinite sets, one way to count their members is to put them into one-to-one correspondence with some standard set, like the set of natural numbers or the set of real numbers. These two sets (i.e., the naturals and the reals) have different sizes: it turns out that the set of natural numbers cannot be put into one-to-one correspondence with the real numbers. No matter how one tries to do it, there will be a real number that has been left out. In this sense, there are “more” real numbers than natural numbers, even though both sets are infinite.
Thus, a useful classification for infinite sets is as “countable” (can be put into one-to-one correspondence with the naturals) or “uncountable” (too big to be put into one-to-one correspondence with the naturals). The rational numbers are countable, so any infinite subset of rationals is also countable. When CronoDAS says that there are as many rationals between zero and X as there are greater than X, he means that both such sets are countable.