> It matters whether you are asking "how many" (cardinals) or "in what position" (ordinals)
But "even" and "odd" are all about whether you can partition something into an equal number of pairs or not. If you're asking "in what position" (ordinals), you've explicitly said you're not in the realm of counting sets of things. I would argue division makes no sense in the realm of ordinals! Everyone is saying the transfinite ordinals alternate even-odd, but those are exactly the numbers where we've stated we're only interested in position, not counting. It's not clear to me why "dividing" an ordinal number into equal pairs makes any sense. (Whereas it makes perfect sense for cardinal numbers.)
> But "even" and "odd" are all about whether you can partition something into an equal number of pairs or not.
Sez you. I can just as easily define even and odd in terms of whether or not I can arrive at a given position in a (potentially infinite) sequence taking by taking two steps at a time.
But "even" and "odd" are all about whether you can partition something into an equal number of pairs or not. If you're asking "in what position" (ordinals), you've explicitly said you're not in the realm of counting sets of things. I would argue division makes no sense in the realm of ordinals! Everyone is saying the transfinite ordinals alternate even-odd, but those are exactly the numbers where we've stated we're only interested in position, not counting. It's not clear to me why "dividing" an ordinal number into equal pairs makes any sense. (Whereas it makes perfect sense for cardinal numbers.)