What do you mean different infinite settings? Do you mean climbing up the 'infinity' ladder by taking power sets of infinite sets? Anyway, I was reluctant on using countables and uncountables because I didn't want to keep introducing new concepts.
Yeah, you're correct about CH, that's embarrassing, I edited the article accordingly. Thank you.
I agree that discussing the general concept of infinity (by which I mean ordinals) is unnecessary for an introduction. I meant to say that if you are only discussing countable and uncountable then you can directly start with this as opposed to starting with infinity which is much broader concept. But this is a matter of taste I think.
Hmm, I guess you're right. It might have been better to center this whole article about countability and uncountability, but obviously, at some point, I got carried away with injections and bijections that I kind of lost the chance to introduce it, I was rushing to get to diagonalization, I really like it lol. This isn't exactly friendly to non-mathematicians I guess.