[yjk]

I think the reals are also even: If x is rational pair it as you would in the rational case (which we assume is even - I haven't proven this). Otherwise pair it to -x, and thus the reals are even.

Being "even" seems like a much more interesting (and simpler) property of a set. I don't see what use there could be to know that you could pair things off, with one element left over. When you extend the notion you do have to decide what to preserve, but to me parity is much more about divisibility and symmetry than it is about reminader. I agree that it's arbitrary, though less arbitrary than the odd definition.

[crazygringo]

If you want to pair positives with negatives, then reals would still be odd, as long as zero is unsigned. Zero is the unpaired element, hence odd.

But it all just seems silly. We can say the set of positive integers is even because we can come up with a pairing of elements, while the set of positive reals is odd because we can't come up with a pairing? Where's the mathematical utility in that?