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, ...]
We can infinitely extend this 1-1 correspondence between these two disjoint subsets, so omega + 1 is evenly divisible. (Or, again, it can also be odd if you choose to arrange the elements differently.)
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).
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:
We can infinitely extend this 1-1 correspondence between these two disjoint subsets, so omega + 1 is evenly divisible. (Or, again, it can also be odd if you choose to arrange the elements differently.)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).