Hm, I'm not so sure that just because the definition of a bijection is technical that it is not intuitive.
I'll start an enumeration of the rationals
1 1/3
2 1/4
3 1/5
...
If you can prove that you can do this, is that really so non-colloquial? It is certainly what we mean by "as many" for all finite sets, so what is wrong with doing this for infinitely many sets?
If every whole positive number is a fraction, but not every fraction is a whole positive number, then colloquially, I wouldn't define them as having "as many" elements as each other. Now, if you want to say they have the same cardinality (and you define cardinality as existing a bijection), then I would agree fully.
Wouldn't there be exactly twice (or twice + 1, if you allow negative fractions) as much fractions, since fractions are represented as two positive numbers (plus a bit if you consider the sign).
(The encoding could be "represent both numbers in binary, put the denominator in the odd bits (LSB = first bit), and the numerator in the even bits" so 2/3 => 10/11 => 1110 => 14)
The thing is that you could also use this kind of logic to show that there are more natural numbers than there are natural numbers. For example, you could associate 1 with all of the numbers from 1 to one million, and still have enough numnbers 'left' to associate each of 2,3,... with distinct natural numbers above one million.
I'll start an enumeration of the rationals
1 1/3
2 1/4
3 1/5
...
If you can prove that you can do this, is that really so non-colloquial? It is certainly what we mean by "as many" for all finite sets, so what is wrong with doing this for infinitely many sets?