>>11731580>I think it's cool because many people say CH is "intuitively true"I think that depends on the formulation. It's "infinitely true" in the "there's no set between, in size, N and R" and that's because people can't imagine well the there object that actually have no tangible counting functions. I.e. indeed people can't even properly "see" the subsets of N. In this formulation, CH is about a "weird not before seen inbetween" objects between the set N they know and between the set R they mistakenly think they know. Because of this mistake, people have the intuition that R is the smallest uncountable set.
But if you abstractly define ordinals, then |omega_1| suddenly becomes the abstractly defined smallest uncountable set. That is, the intuition people tie to R is realized in formal sense. And now CH is actually the claim that R is small, namely as small as just omega_1 .
So in that formulation, to me at least, there's no clear reason why CH is true.
Can we make parts the example more explicit?
Looks like we can choose to have for . That is to say, we can use all finite ordinals to cover the dense rational subset of [0,1] . At this point we make a first limit ordinal jump in induction, jumping beyond .
Of course, we can choose some field extension of Q, say by , count it and have map onto those. With this new objects we get to . And so on.
Adopting CH says that the remaining subset of [0,1] beyond the rational, algebraic etc. is actually "so sparse" that the first jump beyond countable ordinals exhausts it.
R is uncountable by Cantors diagonalization.
When it's possible that an eventual jump to doesn't exhaust the reals in [0,1], then at there's still reals left, and this then has to be an uncountable subset of [0,1].
