There is a famous passage on the third page of Deligne's second paper on the Weil conjectures where he expresses his dislike of the axiom of choice, as manifested in the isomorphism between $C$$\mathbb{C}$ and $\bar{Q_p}$$\overline{\mathbb{Q}}_p$. The proof of said isomorphism runs as follows. Both $C$$\mathbb{C}$ and $\overline{Q_p}$$\overline{\mathbb{Q}}_p$ have transcendence bases, $S$ and $T$. Then $C\simeq \overline{Q(S)}$$\mathbb{C}\simeq \overline{\mathbb{Q}(S)}$ and $\overline{Q_p}\simeq \overline{Q(T)}$$\overline{\mathbb{Q}}_p\simeq \overline{\mathbb{Q}(T)}$. But $C$$\mathbb{C}$ and $\overline{Q_p}$$\overline{\mathbb{Q}}_p$ have the same cardinality, and hence, so do $S$ and $T$. Therefore, $Q(S)\simeq Q(T)$$\mathbb{Q}(S)\simeq \mathbb{Q}(T)$ and, from there, $C\simeq \overline{Q_p}$$\mathbb{C}\simeq \overline{\mathbb{Q}}_p$.
For myself, this proof is quite convincing. Recently, Torsten Ekedahl expressed his opinion to the contrary, and this led to the following exchange:
Why worry about the axiom of choice?
So I wondered about other expert opinions on this matter. Do you find the isomorphism unbelievable and, if so, why?
