posted by Bob .
If there are as many rational numbers as there are irrational numbers then the set of all irrational numbers is infinite
It is true that both sets are infinite. However, it is also true that the set of irrational numbers is so much larger than the set of rational numbers, that if you throw a dart at the number line, the probability of hitting a rational number is zero!
Do a web search on transfinite cardinals to see that the number of rational numbers (and also the number of integers, which is the same!) is the "smallest" infinity, in a sense.
The number of real numbers (or irrationals, which is the same!) is the "next larger" infinity.