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George Cantor proved that the cardinality of $\mathbf{c}$ is larger than the smallest infinity, $\aleph_0$. And he proved that $\mathbf{c}$ equals $2^{\aleph_0}$.

Im looking for the actual paper(s) he wrote down these proofs (in english). Anybody know where and if they are freely downloadable somewhere? If its a simple proof I can accept it as answer, but I really like to have the original paper.

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    Re: your title, that's not a proof of the continuum hypothesis. The continuum hypothesis says that $2^{\aleph_0}=\aleph_1$, that is, that there is no set of cardinality intermediate between $\aleph_0$ and $2^{\aleph_0}$. The proof that $\mathfrak{c}=2^{\aleph_0}$ is basically trivial. – Noah Schweber May 11 '20 at 15:12
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    This is not the continuum hypothesis. – Andrés E. Caicedo May 11 '20 at 15:12
  • Ok my bad. Fixed it. –  May 11 '20 at 15:19
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    Actually, it's not fair for me to say that $\mathfrak{c}=2^{\aleph_0}$ is basically trivial; while the idea is basically trivial ("look at binary expansions of reals"), the detailed implementation is a bit messy. See e.g. here. – Noah Schweber May 11 '20 at 15:19
  • @NoahSchweber Well, with Cantor-Bernstein, it is basically trivial. – Jean-Claude Arbaut May 11 '20 at 15:25
  • @Jean-ClaudeArbaut But I wouldn't call Cantor-Bernstein itself trivial, and it does take a little bit of tedium to actually get the injections totally correct (which to someone first learning the material may pose a momentary stumbling point). Besides, whipping up an explicit bijection builds character. :P – Noah Schweber May 11 '20 at 15:27

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The wikipedia page gives the references for Cantor's original proof and later diagonal proof of the uncountability of the reals. I believe an English translation of the paper containing the diagonal argument can be found in the collection "From Immanuel Kant to David Hilbert: A Source Book in the Foundations of Mathematics, Volume 2."

As to the proof that $\mathfrak{c}=2^{\aleph_0}$, or less symbol-y that there is a bijection $\mathbb{R}$ and $\mathcal{P}(\mathbb{N})$, this may be folklore since it's fairly simple and doesn't involve a big new idea. See e.g. here.

That said, I would not recommend the original papers for these, or any other, basic results in logic since subsequent texts provide much clearer explanations (unless your interest is historical).

Noah Schweber
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  • Since it is saying something about the reals in relation to the natural numbers, cardinality of the power set of the natural numbers ($2^{\aleph_0}$). I like to know a little bit about the history, since I am studying the basis notation in binary expansion using $2^b$ where $b\in{0,1}$. Since $\mathbb{R}$ and $\mathcal{P}(\mathbb{N})$ are equivalent, Im just trying to get some connections to the Collatz conjecture where the question is wether any initial number approaches $2^n$. Id like to study these kinds of proofs because they are hard to find in the massive literature out there. –  May 11 '20 at 15:53
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    @Natural Number Guy: I like to know a little bit about the history --- See my answer to Where was it first proved that the cardinality of the continuum equals the cardinality of the power set of the naturals? Regarding I am studying the basis notation in binary expansion, the concept of binary and other notational systems was a common topic in 1800s school-level (by present standards) algebra texts under the name scales of notation. – Dave L. Renfro May 11 '20 at 17:21