Let $\mathscr{S}$ be the set of all countable subfields of $\mathbb{C}$. Clearly, $\mathscr{S}$ is a partially ordered set under inclusion, and if $K_1\subseteq K_2 \subseteq \cdots$ is an ascending chain of countable subfields, then $\bigcup_{i=1}^{\infty}K_i$ is a countable union of countable fields, and is hence an upper bound for $K_1\subseteq K_2 \subseteq \cdots$ which is in $\mathscr{S}$. But then $\mathscr{S}$ satisfies the conditions of Zorn's lemma, so there is some maximal element $K$. It would then seem that $K$ is a countable field such that whenever $K\subsetneq L\subseteq \mathbb{C}$ is a pair of field extensions, we have that $L$ is uncountable. This seems quite unintuitive to me. Has anyone exhibited such a subfield of $\mathbb{C}$ and proved that it has the properties required? It would seem that for any countable subfield $K\subseteq \mathbb{C}$, there will be some complex number $\alpha\notin K$, in which case $K\subsetneq K(\alpha)\subset\mathbb{C}$ and $K(\alpha)$ is also countable.