But that seems redundant with countable/uncountable sets, because then every countable infinite set would be even (e.g. rational numbers), and every uncountable infinite set would be odd (e.g. real numbers).

It's also not clear to me what justification there would be for a "preference" for the "even" category that way -- it seems arbitrary. Why not be odd if there exists a way to pair things off such that one is left over, and even if there isn't such a way?

Because evenness is a special case of k-evenness: A set is k-even if it can be divided into equal sets of size k. Which, for finite sets, is equivalent to the size of the set being 0 mod k, ie, is divisible by k. There are many ways to be not be divisible by any particular number bigger than two, and only one way to be divisible.

Uncountability is a particularly interesting form of non-divisibility, so I'm just fine calling all uncountable sets odd and countable sets even...

(And just because we're hung up on divisibility by two, let us remember: All prime numbers are odd, and two is the oddest of them all.)

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.

If you can pair but not count, the reals would be odd I think, as long as zero is unsigned. Zero would make it odd.

But pairity is just a question of sorting the countable numbers into sets of size two, and the more general form even of that is sorting into sets of size N. It's just as easy to say that the countable numbers are odd if there exists a way to sort them into sets of size three. So I'd argue the countable numbers are odd.

And you then you could retort with sets of size four, and I could use five, and then we can argue about whether we'll end up at the limit with more odd sets or even sets, and now we're arguing in circles. Reductio ad absurdum.

Why? You can group 30 into sets of 3 (3 x 10), but 30 is still even, so your definition of odd doesn't hold.

Indeed, parity is the more general form of even/oddness