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.
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).