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.
Modern mathematics is all about coming up with definitions and rules that give rise to interesting (to a mathematician!) properties when further investigated.
The definition given naturally lets the ordinal numbers continue the odd/even/odd... pattern. Choosing the alternative definition would not.
In one sense that's 'arbitrary' because we decided on one definition over another. But another sense, we picked the parity rule that lets us extend the same pattern from the natural numbers, so it's a 'better' parity rule. And the fact that one rule gives this pattern while the other does not, did not come from humans, but is a 'metamathematical fact' from the universe of possible ways to define things.
So I would say this definition is not fully arbitrary, it's an interaction between what mathematicians find interesting and the Platonic realm of possible mathematical constructs.
Anyway, I'm not a mathematician but it seems this is how the game of math is played: to continually discover new rules that give rise to more interesting math.
So there is no better parity rule as you say, it is entirely arbitrary. It's not extending the same pattern, it's seeing that there are two ways of extending it and picking one arbitrarily.
In that case, if it were me, I'd call it something else. Not parity or even/odd. Because it's not the natural obvious extension. It's arbitrary.
To get a feel for why this is convenient, consider that you can generalize by replacing "multiples of 2" with "multiples of n". Then, instead of splitting everything into two sets (even/odd), we can naturally split the integers into n sets called equivalence classes modulo n. For n=10, these would be "multiples of 10", "numbers whose remainder after dividing by 10 is 1", "numbers whose remainder after dividing by 10 is 2", and so on. Seen this way, you may find it less arbitrary now.
There are just as many odd numbers as even, so there's nothing more natural about it. They alternate. Yes you can extend to higher multiples, but there's still nothing more natural about multiples of 7 vs. multiples of 7 with remainder 3.
And it's just as easy to say that infinity is divisible by 7, as it is to say that it divisible by 7 with remainder 3:
[1, 2, 3, 4, 5, 6, 7], [8, 9, 10, 11, 12, 13, 14], ...
1, 2, 3, [4, 5, 6, 7, 8, 9, 10], [11, 12, 13, 14, 15 16, 17], ...
Still, there's a good reason to privilege the multiples. With regular addition of the integers, the number zero has a special role, in that n + 0 = 0 + n = n for all n. It's called the "additive identity", and it's the only number that has this property. If we think of inverses of numbers, like "what's the opposite of 19?", then in the world of addition, they are defined in relation to 0. The "opposite" of 19 is -19, because 19 + (-19) = 0.
Many algebraic structures have an identity; in the world of multiplication of fractions, the identity is 1, and the inverse of 19 is now 1/19. A more abstract example would be the operations on a Rubik's Cube, where the identity is "do nothing". That's the least exciting thing to do with a Rubik's Cube, but it has a special role, just like 0 with addition. If we want to talk about inverses of Rubik's operations, then again, they are defined in relation to the identity: the opposite of "rotate the top face a quarter turn clockwise" is "rotate the top face a quarter turn counterclockwise", because the sequence of those two operations gives you "do nothing".
It is in this sense that "multiples of n" are special, because they effectively comprise the identity element under addition modulo n. That is, if we add numbers and only look at the last digit (in other words, the remainder after dividing by 10), we'll find that adding 0, or 10, 20, 30, etc., leaves that digit unchanged. Another way to say this is that if you take two numbers with the same last digit, their difference will be a multiple of 10.
In other words, it isn't merely that there are just as many numbers in one set as another, it's that one of the sets acts as a point of reference. For a real-world metaphor, consider the concept of birthdays (disregarding complications like leap years). If you were born on February 5, then every other February 5 is a birthday, because the difference of those two dates is a multiple of 365. This might highlight the conceptual argument: I would agree that there's nothing fundamentally more special or interesting about February 5 than August 27 or any other day, but it's when we start comparing dates or using them in some frame of reference (like trips around the sun) that the number 365 and its multiples come into focus.
Or, for a real-world example related to evenness vs. oddness, go and flick a light switch an even number of times. If the light was off to begin with, it will still be off at the end; if it was on, it will still be on. Now, if you have a fancy lamp with three settings, then turn the switch a multiple of 3 times. Again, this will preserve the state, and this is why multiples are in some sense special.
Finally, as for infinity: I'm with you in that it gets a bit uncomfortable to talk about the evenness or oddness of infinity itself. At that point it really comes down to the choice of definitions, and a perfectly reasonable definition is that infinity isn't a number but an unattainable goal (it's the trip, not the destination), in which case the concepts of evenness and oddness don't apply at all.
I'm arguing that because it's just as easy to say that omega is odd as to say that it's even, that the whole concept breaks down and loses and all meaning.
Because if you want to divide omega + 1 in half to show that it's even, we can do that. If we denote the set element inside of the "1" of "+ 1" by the symbol "a", then we can write out:
[1, 3, 5, 7, ...]
[a, 2, 4, 6, ...]
But I'm not saying that this is useful or interesting. My whole point is that it's not because even/odd is not meaningful at all for transfinite numbers, because they're just as odd as even. That in the same way there's no utility in attempting to decide whether the decimal 2.7 is odd or even, there's similarly no utility in defining omega as odd or even (or omega + 1).
Actually the fact that splitting it into pairs is the same as splitting into two equal sets of equal cardinality is itself non-trivial. The reason why shows up when you try to get the two definitions closer together.
Splitting an ordinal into pairs is essentially splitting it into ordered pairs (a_i, b_i) such that the map i to a_i is monotonic and for no i<j the pair (a_i, b_i) overlaps with (a_j, b_j) in the sense that a_j <= b_j.
Splitting a set into pairs is splitting it into sets {a_i, b_i} such that for no i != j the two sets {a_i, b_i} and {a_j, b_j} overlap.
These two are note the same, you can split pretty much any infinite set into two disjoint sets of equal cardinality.
It's hard to get the definitions general enough to get one definition for both ordinals and sets. Mostly because products of ordinals are a bit weird. For sets (and most other types of mathematical objects) it doesn't matter which way around you pair things up, but for the ordinals you end up with a completely different object if you do it the other way around and this is apparently the more interesting definition of the two.
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?
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.)
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.
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.
It's only even or odd with respect to other infinities which the cardinal numbers can count based on the presence of a bijection or not. It's a kind of relative parity.
That directly contradicts the quoted text I included from the original answer though, as far as I understand. It directly asserted that "every infinite cardinal is even".
> It's a kind of relative parity.
What is relative parity? The original question was whether infinity is even or odd... I don't know what you mean by relative parity.
> maybe I'm missing something
The answer said:
> the usual definition is that an ordinal number 𝛼 is even if... Otherwise, it is odd.
In other words, if a number could be proved to be even, it is even. If not, it is odd.
Using their definition, there is no such thing as "proving a number is odd". You'd have to do it by failing to prove it's evenness. In the case of infinity, because we can successfully prove evenness, it's even and not odd.
> 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.)