TITLE: Proofs of the uncountability of the reals
QUESTION [92 upvotes]: Recently, I learnt in my analysis class the proof of the uncountability of the reals via the Nested Interval Theorem (Wayback Machine). At first, I was excited to see a variant proof (as it did not use the diagonal argument explicitly). However, as time passed, I began to see that the proof was just the old one veiled under new terminology. So, till now I believe that any proof of the uncountability of the reals must use Cantor's diagonal argument.
Is my belief justified?
Thank you.

REPLY [4 votes]: Today I attended a lecture, and this question was asked, after the talk, I realized the following proof, that might be interesting, at least for set theorists!!!:

Theorem (Cantor): The set of real numbers is uncountable.
Proof: Suppose not. Let $M$ be a countable transitive model of enough of ZFC such that $\mathbb R\in M$.
By our assumption $\mathbb R\subseteq M$.
Let $\mathbb P=\operatorname{Add}(\omega,1)$ be the Cohen forcing for adding a new real, let $G\subseteq\mathbb P$ be generic over $M$ and let $r=\bigcup G$.
Then:

$r\notin M$, as otherwise $G=\{r \restriction n: n<\omega  \}\in M$ which is not possible
$r\in\mathbb R\subseteq M$

A contradiction.