## Same but multifaceted July 11, 2008

Posted by Alexandre Borovik in Uncategorized.

Continuing the topic of “sameness”, it is interesting to compare behaviour of two familiar objects: the field of real numbers $\mathbb{R}$ and the field of complex numbers $\mathbb{C}$.

$\mathbb{C}$ 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 $2^{\aleph_0}$.

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 $2^{2^{\aleph_0}}$ (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. I am still hunting appropriate references for this fact. Meanwhile, I got the following e-mail from a model theorist colleague, Zoe Chatzidakis:

Models of uncountably categorical theories behave really like vector spaces: if inside a model $M$ you take a maximal independent set $X$ of elements realizing the generic type, and take any permutation of $X$, it extends to an automorphism of the model. So, if $M$ is of size $\kappa > \aleph_0$, then any basis has size $\kappa$, and its automorphism group has size $2^\kappa$.

I don’t know a reference, but it should be in any model theory book which talks about strongly minimal sets. Or maybe in the paper by ??? Morley ??? which shows that you have a notion of dimension and so on? I.e., that $\aleph_1$ categorical theories and strongly minimal sets are the same.

It is really a well-known result, so you probably don’t need a reference if you cite it in a paper.

1. dcorfield - July 21, 2008

There’s interesting work being done on what’s special about the complex numbers that they feature in quantum mechanics. See, for example, Jamie Vicary’s Categorical properties of the complex numbers.

2. anon - July 24, 2008

There is an old article by Paul Yale on JSTOR that is easy to find; google “automorphisms complex numbers”, and it’s the first link. It explains these “wild automorphisms” of the complex numbers in quite a friendly way.

3. dcorfield - July 25, 2008

Great, thanks!

Gosh, if $\phi$ is a wild automorphism of $\mathbb{C}$ then $\phi$ is a discontinuous mapping of the complex plane onto itself; in fact, $\phi$ leaves a dense subset of the real line pointwise fixed but maps the real line onto a dense subset of the plane.

4. John G. - August 1, 2008

Some of the seminal research that Zoe was alluding to appeared in Michael Morley’s Ph.D. thesis, and some of it appeared in Baldwin and Lachlan’s 1971 paper “On strongly minimal sets” (Journal of Symbolic Logic, vol. 36, pp. 79-96).

As she said, these sorts of things are now folklore in model theory circles. The work by Morley and Baldwin-Lachlan is actually quite accessible, but there are also many nice modern presentations, e.g. in Hodges’ textbook “A Shorter Model Theory” (see the final chapter on “Structure and categoricity”).

Your intuition about “multifacetedness” can be made more precise: it’s well-known that any uncountable model of an uncountably categorical theory is saturated and homogeneous (as defined in any beginning model theory textbook). Actually this was proved in Morley’s thesis, too.

5. Alexandre Borovik - August 2, 2008

my warmest thanks!,

6. Seamus - August 11, 2008

I’m only really finding my feet in model theory, but how does all this fit with the idea that you can think of complex numbers as ordered pairs of reals?