David Corfield
The Complex Numbers

\mathbb{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 \mathbb{R}, its elementary theory, that is, the set of all closed first order formulae that are true in \mathbb{R}, has infinitely many models of cardinality continuum 2 02^{\aleph_0}.

In naive terms, \mathbb{C} is rigid, while \mathbb{R} is soft and spongy and shape-shifting. However, \mathbb{R} has only trivial automorphisms (an easy exercise), while \mathbb{C} has huge automorphism group, of cardinality 2 2 02^{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 \mathbb{R}, while \mathbb{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 \mathbb{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.

Last revised on April 28, 2009 at 19:30:33. See the history of this page for a list of all contributions to it.