David Corfield
The Complex Numbers

• Sasha

$ℂ$ is uncountably categorical, that is, it is uniquely described in a language of first order logic among the fields of the same cardinality.

In case of $ℝ$, its elementary theory, that is, the set of all closed first order formulae that are true in $ℝ$, has infinitely many models of cardinality continuum $2^{{\aleph }_0}$.

In naive terms, $ℂ$ is rigid, while $ℝ$ is soft and spongy and shape-shifting. However, $ℝ$ has only trivial automorphisms (an easy exercise), while $ℂ$ has huge automorphism group, of cardinality $2^{2^{{\aleph }_0}}$ (this also follows with relative ease from basic properties of algebraically closed fields). In naive terms, this means that there is only one way to look at $ℝ$, while $ℂ$ can be viewed from an incomprehensible variety of different point of view, most of them absolutely transcendental. Actually, there are just two comprehensible automorphisms of $ℂ$: the identity automorphism and complex conjugation. It looks like construction of all other automorphisms involves the Axiom of Choice. When one looks at what happens at model-theoretic level, it appears that “uniqueness” and “canonicity” of a uncountable structure is directly linked to its multifacetedness.