Though an interesting discussion ...
My 2 cents
- an element can only be in a set once;
- e.g. 1/2 and 2/4 are different representations of the same number; so ...
- depending on the perspective (form vs. value) they can either both be separate members of a set, or they must be considered the same member;
- I lean towards considering the Rationals a set of values, but I fully understand the alternative perspective;
- In the end, what counts (granted, a corny pun) is that either way the Rationals are countable; it doesn't matter by which index number each Rational is referred to.
... it's off-topic. 