OK, so I guess I'm just understanding that mathematicians arbitrarily decided to prioritize "even" over "odd"?
Because as I stated in another comment, you could just as easily say odd cardinality exists if you can find two subsets with the same cardinality and there's one element left over, and otherwise we call it even.
So at the end of the day, what you're saying is that ultimately infinity would be even just because mathematicians arbitrarily defined 'even' that way -- not because there's any intuitive logic behind it, any deeper justification, or any necessary consistency with parity for finite sets.
Because as I stated in another comment, you could just as easily say odd cardinality exists if you can find two subsets with the same cardinality and there's one element left over, and otherwise we call it even.
So at the end of the day, what you're saying is that ultimately infinity would be even just because mathematicians arbitrarily defined 'even' that way -- not because there's any intuitive logic behind it, any deeper justification, or any necessary consistency with parity for finite sets.