[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.