Newbie approach to understand generalized continuum hypothesis

Question

There is this theorem that size of power set constructed from infinite set is "more" infinite than the previous set:

$$ \begin{eqnarray*} \aleph_0 &= |\mathbb{N}| \\ \aleph_{n+1} &= 2^{\aleph_n} \\ \end{eqnarray*} $$

I always understood the real numbers as $(a,b) \in \mathbb{N} \times \mathbb{Z}$, where $a \in \mathbb{N}$ is the part "prior" to comma, and the $b \in \mathbb{Z}$ is part "past" the comma. This seems to make it possible to construct any numbers, be it rational, irrational or whatever. (Correct me if I'm wrong.) This kind of reasoning, however, seems to clash with GCH. Namely, while $|\mathbb{N}| = |\mathbb{Z}| = \aleph_0$, my reasoning implies that $\mathbb{N}^2 \equiv \mathbb{R}$, which according to GCH is false, because $\aleph_1 \neq \aleph_0^2$ and $\aleph_1 = 2^{\aleph_0}$.

I try to imagine real numbers as a power set of natural numbers, but I fail.

My question is: why my reasoning that $\mathbb{R} \equiv \mathbb{N}^2$ is wrong? And how to comprehend the idea that $2^\mathbb{N} \equiv \mathbb{R}$, i.e. how to use power set of natural numbers to obtain/construct real numbers?

As a hobbyist I'd like illustrative, practical examples the most instead of going into raw stuff.

Answer 1

There is this theorem that size of power set constructed from infinite set is "more" infinite than the previous set.

Yes, its called Cantor's theorem. But it doesn't say that $2^{\aleph_\alpha} = \aleph_{\alpha+1}$. That's a (famous) strictly stronger statement called GCH, and it is independent of the usual axioms of set theory (most set theorists believe that if there is a true universe of sets, then GCH is probably false in that universe. However, the usual axioms can neither prove nor refute GCH.)

Cantor's theorem is the weaker statement that $2^{\aleph_\alpha} > \aleph_\alpha$. Actually, it really has nothing to do with the $\aleph$ numbers, so the best statement of Cantor's theorem is surely: "$2^\kappa>\kappa$ for any cardinal number $\kappa$." Contrary to GCH, Cantor's theorem is more-or-less uncontroversial, except among certain laypeople.

I always understood the real numbers as $(a,b)∈\mathbb{N×Z}$, where $a∈\mathbb{N}$ is the part "prior" to comma, and the $b∈\mathbb{Z}$ is part "past" the comma.

This makes no sense to me. Are you sure you're not mixing up the real numbers with the rational numbers? I often think of $\mathbb{Q}$ as being a quotient of $\mathbb{(N\setminus\{0\})×Z}.$

My reasoning implies that $\mathbb{N}^2≡\mathbb{R}$.

As you correctly surmise, this is incorrect, by Cantor's theorem. If you were to prove the truth of $\mathbb{N}^2≡\mathbb{R}$, you would prove the entirety of modern set theory logically inconsistent. (Needless to say, I don't think this is very likely.)

My question is: why my reasoning that $\mathbb{N}^2≡\mathbb{R}$ is wrong?

Well we'd be happy to play with "find the error" game with your proof (this can be quite an instructive exercise), but you haven't really given us a proof!

My verdict. This site works best when you have already have a basic understanding of how things "fit together" in the branch of math you're asking questions about. Since your question is predicated on more than one misconception, I think you should find yourself a good book to learn the basics, even if you're just an interested hobbyist. Trying to learn the basics purely by reading wikipedia and/or asking question at math.stackexchange is a crappy approach to learning the fundamentals; trust me, I've tried it at least twice.

I'd recommend Goldrei's set theory for a readable introduction. If you work through an introductory set theory book (asking questions on this site as they arise, of course), things will be a lot clearer, and the site will be more useful to you.