1. Between every two rational numbers are an infinite number of irrational numbers.

I agree with this. I understand that Cantor showed that the set of rational numbers (=fractions) is "countable": you can define a 1:1 relationship between all fractions and the whole numbers.

However, the set of irrational numbers is uncountable, ie it is infinitely more than the set of rational numbers.

2.Between every two irrational numbers are an infinite number of rational numbers.

I don't see how this is consistent with (1). If you pick any two of the infinite set of irrational numbers "between" two "closest" rational numbers, then is this a counterexample?

For how to count by infinities, see:

http://en.wikipedia.org/wiki/Aleph_number