### Real versus Complex Numbers

#### Posted by David Corfield

Over here, we’re looking at the differences between two famous infinitely large fields:

$\mathbb{C}$ 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.

Apparently, there is something Galoisianly model theoretic going on.

Now, I see Jamie Vicary has a new paper – Categorical properties of the complex numbers, and I am wondering whether bridges can be built.

Abstract:

Given the success of categorical approaches to quantum theory, and given that quantum theory is underpinned by the complex numbers, it is interesting to study the categorical properties of the complex numbers directly. We describe natural categorical conditions under which the scalars of a monoidal $\dagger$-category gain many of the features of the complex numbers. Central to our approach is the requirement that the $\dagger$-functor be compatible with the construction of particular limits in the category; we show that this implies nondegeneracy of the $\dagger$-functor, as well as cancellable hom-set addition. Our main theorem is that in a nontrivial monoidal $\dagger$-category with finite $\dagger$-biproducts and finite $\dagger$-equalisers, for which the monoidal unit has no proper $\dagger$-subobjects, the scalars have an involution-preserving embedding into an involutive field with characteristic 0 and orderable fixed field, and therefore embed into the complex numbers if they are at most of continuum cardinality.

In the discussion he writes,

We set out to find a set of properties of a category that imply the category contains an algebraic structure similar to the complex numbers. We have achieved this, to some extent. The most significant missing property is that, although we have shown that the self-adjoint scalars admit a total order, this will not necessarily resemble the order on the real numbers. As a consequence, although the scalars will embed into the complex numbers (as long as they are at most of continuum cardinality), we cannot guarantee that an embedding can be found that is involution-preserving, sending the self-adjoint scalars into the real numbers.

The problem seems to be that while the reals can be characterised as

the ‘biggest’ Archimedean order: for every field with a specified Archimedean order, there is an order-preserving embedding into the real numbers. In particular, any such field is at most of continuum cardinality. We know that our self-adjoint scalars will admit an order, but it is possible that they will fail to admit an Archimedean order, which is a necessary property for them to admit an embedding into the real numbers.

## Re: Real versus Complex Numbers

This confused me. (It confused me over at your other blog too.)

Algebraically closed fields of characteristic zero are uncountably categorical, right? So it’s nothing particular about C. And from that point of view, I can explicitly construct other automorphisms. For example, I can take the algebraic closure of the p-adics; I can probably find an explicit element of the Galois group there. (I don’t actually know that’s the case, but I’m guessing it’s true.)

What makes C special is that its constructed from R. So in this argument, you are explicitly choosing your first-order model of R, i.e. R itself. You could take one of the other models, construct the equivalent of C over that model, and viola you have another algebraically closed field of characteristic zero. This field is isomorphic to C, but I would guess that you would need the axiom of choice to prove it.