[kccqzy]

Once you get to infinities, you can no longer denote the "size" of a set using naturals numbers, which is the typical usage of the word "size" (there are three apples in my basket and three is a natural number).

So to me this is just quibbling about the definition of the word "size" which isn't a productive conversation. Stop calling it "size" and give it a specific terminology ("cardinality") instead and the whole unintuitive naming problem is sidestepped.

[usrusr]

You can even give in a little more without accepting the blatant takeover of "bigger": accept that there are different qualities in infinite comparisons. A difference in cardinality can certainly be considered a stronger difference than everything mappable, but denying infinites like "number of points on an infinite line" it's smaller-than relative to "number of points on two infinite lines" is just ridiculous. It's snobbish, pure vanity.

[kccqzy]

My point is that intuitive but imprecise naming misleads people. There's no way to talk about "number of points on an infinite line" because there isn't a specific natural number that you can find to denote the number of points here. Infinity is not a number.

That's why we need a proper concept of cardinality.