So there seems like a glaring hole in the answer, but maybe I'm missing something. Because:
> It is easy to prove from this definition by transfinite recursion that the ordinals come in an alternating even/odd pattern, and that every limit ordinal (and hence every infinite cardinal) is even.
Sure, if we use the natural numbers and start at 1, then we can group:
[1, 2], [3, 4], [5, 6], ...
and prove infinity is even.
But we could also just as easily group:
1, [2, 3], [4, 5], [6, 7], ...
and prove infinity is odd.
It's the same if we try to split into two equal subsets, because we can split into:
[1, 3, 5, ...]
[2, 4, 6, ...]
and say it's even. Or we can divide:
1
[2, 4, 6, ...]
[3, 5, 7, ...]
and prove it's odd because we have two equal subsets plus one left over.
So I'm missing the reason for why the second versions aren't just as valid.
(Of course, I'm more inclined to agree with many commenters here that it's just a category error, and asking whether infinity is even/odd is as useful as asking whether democracy is blonde or brunette.)
> It is easy to prove from this definition by transfinite recursion that the ordinals come in an alternating even/odd pattern, and that every limit ordinal (and hence every infinite cardinal) is even.
Sure, if we use the natural numbers and start at 1, then we can group:
and prove infinity is even.But we could also just as easily group:
and prove infinity is odd.It's the same if we try to split into two equal subsets, because we can split into:
and say it's even. Or we can divide: and prove it's odd because we have two equal subsets plus one left over.So I'm missing the reason for why the second versions aren't just as valid.
(Of course, I'm more inclined to agree with many commenters here that it's just a category error, and asking whether infinity is even/odd is as useful as asking whether democracy is blonde or brunette.)