A wonderful mistake, which paved the way to singular cardinals, was done by Felix Bernstein in his dissertation. I learnt this from Menachem Kojman. Berstein thought he had proved that for every ordinal $\alpha$, $\aleph_\alpha^\omega=\aleph_\alpha \cdot 2^{\aleph_0}$. This is true for every $\alpha < \omega$ but already fails for $\alpha=\omega$. Berstein's mistake was to assume that every cardinal has an immediate predecessor.

Kőnig later used Berstein's result to prove that the continuum is not an aleph, thus disproving at once two of Cantor's main beliefs: 1) every set can be well-ordered and 2) the continuum hypothesis! He presented his result at the third International Congress of Mathematicians in Heidelberg in 1904 and the organizers cancelled all parallel session to allow all participants (which included Cantor and Hilbert) to attend Kőnig's lecture. And his discovery was even reported in the local news!

Here is Kőnig's reasoning:

First he proves the correct result that for every ordinal $\beta$, $\aleph_{\beta+\omega}^\omega>\aleph_{\beta+\omega}$ (a special case of what is now known as Kőnig's Theorem). He then reasons that if the continuum were an aleph, say $2^{\aleph_0}=\aleph_\beta$, then substituting $\alpha=\beta+\omega$ into Berstein's result one obtains that $\aleph_{\beta+\omega}^\omega=\aleph_{\beta+\omega} \cdot 2^{\aleph_0}=\aleph_{\beta+\omega}$, which is a contradiction!

A wonderful mistake, which paved the way to singular cardinals, was done by Felix Bernstein in his dissertation. I learnt this from Menachem Kojman. Bernstein thought he had proved that for every ordinal $\alpha$, $\aleph_\alpha^\omega=\aleph_\alpha \cdot 2^{\aleph_0}$. This is true for every $\alpha < \omega$ but already fails for $\alpha=\omega$. Bernstein's mistake was to assume that every cardinal has an immediate predecessor.

Kőnig later used Bernstein's result to prove that the continuum is not an aleph, thus disproving at once two of Cantor's main beliefs: 1) every set can be well-ordered and 2) the continuum hypothesis! He presented his result at the third International Congress of Mathematicians in Heidelberg in 1904 and the organizers cancelled all parallel session to allow all participants (which included Cantor and Hilbert) to attend Kőnig's lecture. And his discovery was even reported in the local news!

Here is Kőnig's reasoning:

First he proves the correct result that for every ordinal $\beta$, $\aleph_{\beta+\omega}^\omega>\aleph_{\beta+\omega}$ (a special case of what is now known as Kőnig's Theorem). He then reasons that if the continuum were an aleph, say $2^{\aleph_0}=\aleph_\beta$, then substituting $\alpha=\beta+\omega$ into Bernstein's result one obtains that $\aleph_{\beta+\omega}^\omega=\aleph_{\beta+\omega} \cdot 2^{\aleph_0}=\aleph_{\beta+\omega}$, which is a contradiction!