An un-odd semantic solution

Last Monday, I reported my "insight" that there seemed to be more even numbers than odd numbers. I labeled the report humor to signal that my tongue was in my cheek. But I also labeled it parody to signal that I was seriously making fun of a certain way of trying to think (exemplified by numerology and astrology and, perhaps more often than not, by theology).
    Commenter Ken provided a serious, logical counter to the "proof" that there are more even numbers than odd. There's an infinite number of both, he pointed out, and one infinity is just as numerous as another. Actually, as Georg Cantor (1845-1918) theorized,

there are infinite sets of different sizes (called cardinalities). For example, the set of integers is countably infinite, while the set of real numbers is uncountably infinite. –Wikipedia

But Ken is right, because, with even and odd numbers, we're confining ourselves to "countably infinite" sets.
    Of course, I never thought for a minute myself that there were more even numbers than odd numbers. But the semantic example I contrived (the asymmetry of "even number of odd numbers" and "odd number of even numbers") did sort of make it look as though there might be more even than odd numbers.
    While Ken's logical objection is well and good¹, I've been waiting for my muse to provide a fitting semantic retort. Alas, I've been disappointed, and I've even had grapefruit almost every morning. I'm going to have to work to provide a proportional solution.

Many years ago (in the early seventies), I became interested in "creative problem-solving." One of the things I learned from my mentor, Moe Edwards ("Moe" was simply his initials; there's a reference on the web to a book titled Doubling Idea Power, by M.O. Edwards, Palo Alto), was to "ask provocative questions"—or to ask any question at all, to see whether it can spur a thought.
    Okay. What's going on when you (1) double an odd number and get an even number, but (2) take an odd number of even numbers and also get an even number?
    What this struck right off is the realization that when you double two things, you get an even number of them, and when you take an odd number of things, you get an odd number of them. So something fishy or sleight-of-hand seems to be going on in (2), taking an odd number [of something]...and getting "an even number."
    And the clue as to what's going on is that the phrase, "of them," was coyly omitted in Monday's "proof." That is, it didn't say "getting an even number of them," but "getting an even number."
    I'll go on and spell this out later (perhaps tomorrow), but for now, as a gift to my readers, I'll leave it to them to work it out for themselves, perhaps while eating a grapefruit.

Proof more odd than even

The proof "more even than odd" was semantic,
Playful, good-humored, a little pedantic,
    Done for good fun,
    As well as the pun,
And while not even odd, it was, flatly, antic².

_______________

1. "Acceptable, all right, as in 'If you can get a better discount elsewhere, well and good.' [The] redundant phrase ['well and good'] was first recorded in 1699." –yourdictionary.com
2. antic. adjective: ludicrously odd
       Example: "Hamlet's assumed antic disposition."