$\begingroup$

If $|\mathbb{C}|=|\mathbb{R}|$ and thus $\mathbb{C}$ is isomorphic to $\mathbb{R}$, why can't $\mathbb{C}$ be totally ordered?

$\endgroup$ 4

1 Answer

$\begingroup$

$\mathbb{C}$ can indeed be totally ordered as a set, since you can find a set bijection to the reals. But for a field the definition of a total order requires that all squares be positive. Since $1$ is positive and $-1$ is negative the fact that $i^2 = -1$ implies that no total order exists.

$\endgroup$