### Uncountably Categorical Theories

#### Posted by John Baez

Right now I’d love to understand something a logician at Oxford tried to explain to me over lunch a while back. His name is Boris Zilber. He’s studying what he informally calls ‘logically perfect’ theories — that is, lists of axioms that almost completely determine the structure they’re trying to describe. He thinks that we could understand physics better if we thought harder about these logically perfect theories:

Boris Zilber, Perfect infinities and finite approximation.

Boris Zilber, On model theory, noncommutative geometry and physics.

His ways of thinking, rooted in model theory, are quite different from anything I’m used to. I feel a bit like Gollum here:

A zeroth approximation to Zilber’s notion of ‘logically perfect theory’ would be a theory in first-order logic that’s **categorical**, meaning all its models are isomorphic. In rough terms, such a theory gives a *full* description of the mathematical structure it’s talking about.

The theory of groups is not categorical, but we don’t mind that, since we all know there are lots of very different groups. Historically speaking, it was much more upsetting to discover that Peano’s axioms of arithmetic, when phrased in first-order logic, are not categorical. Indeed, Gödel’s first incompleteness theorem says there are many statements about natural numbers that can neither be proved nor disproved starting from Peano’s axioms. It follows that for any such statement we can find a model of the Peano axioms in which that statement holds, and also a model in which it does not. So while we may imagine the Peano axioms are talking about ‘the’ natural numbers, this is a false impression. There are many different ‘versions’ of the natural numbers, just as there are many different groups.

The situation is not so bad for the real numbers — at least if we are willing to think about them in a somewhat limited way. There’s a theory of a real closed field: a list of axioms governing the operations $+, \times, 0$ and $1$ and the relation $\le$. Tarski showed this theory is complete. In other words, any sentence phrased in this language can either be proved or disproved starting from the axioms.

Nonetheless, the theory of real closed fields is not categorical: besides the real numbers, there are many ‘nonstandard’ models, such as fields of hyperreal numbers where there are numbers bigger than $1$, $1+1$, $1+1+1$, $1+1+1+1$ and so on.
These models are all **elementarily equivalent**: any sentence that holds in one holds in all the rest. That’s because the theory is complete. But these models are not all isomorphic: we can’t get a bijection between them that preserves $+, \times, 0, 1$ and $\le$.

Indeed, only finite-sized mathematical structures can be ‘nailed down’ up to isomorphism by theories in first-order logic. After all, the **Löwenheim–Skolem theorem** says that if a first-order theory in a countable language has an infinite model, it has at least one model of each infinite cardinality. So, if we’re trying to use this kind of theory to describe an infinitely big mathematical structure, the most we can hope for is that *after we specify its cardinality*, the axioms completely determine it.

And this actually happens sometimes. It happens for the complex numbers! Zilber believes this has something to do with why the complex numbers show up so much in physics. This sounds very implausible at first, but there are some amazing results in logic that one needs to learn before dismissing the idea out of hand.

Say $\kappa$ is some cardinal. A first-order theory describing structure on a single set is called **κ-categorical** if it has a unique model of cardinality $\kappa$. And in 1965, a logician named Michael Morley showed that if a list of axioms is $\kappa$-categorical for *some* uncountable $\kappa$, it’s $\kappa$-categorical for *every* uncountable $\kappa$. I have no idea why this is true. But such theories are called **uncountably categorical**.

A great example is the theory of an algebraically closed field of characteristic zero.

When you think of algebraically closed fields of characteristic zero, the first example that comes to mind is the complex numbers. These have the cardinality of the continuum. But because this theory is uncountably categorical, there is exactly one algebraically closed field of characteristic zero of *each* uncountable cardinality… up to isomorphism.

This implies some interesting things. For example, we can take the complex numbers, throw in an extra element, and let it freely generate a bigger algebraically closed field. It’s ‘bigger’ in the sense that it contains the complex numbers as a proper subset, indeed a subfield. But since it has the same cardinality as the complex numbers, it’s *isomorphic* to the complex numbers!

And then, because this ‘bigger’ field is isomorphic to the complex numbers, we can turn this argument around. We can take the complex numbers, remove a lot of carefully chosen elements, and get a subfield that’s isomorphic to the complex numbers.

Or, if we like, we can take the complex numbers, adjoin a *really huge* set of extra elements, and let them freely generate an algebraically closed field of characteristic zero. The cardinality of this field can be as big as we want. It will be determined up to isomorphism by its cardinality. But it will be elementarily equivalent to the ordinary complex numbers! In other words, all the same sentences written in the language of $+, \times, 0$ and $1$ will hold. See why?

The theory of a real closed field is *not* uncountably categorical. This implies something *really* strange. Besides the ‘usual’ real numbers $\mathbb{R}$ there’s another real closed field $\mathbb{R}'$, not isomorphic to $\mathbb{R}$, with the same cardinality. We can build the complex numbers $\mathbb{C}$ using pairs of real numbers. We can use the same trick to build a field $\mathbb{C}'$ using pairs of guys in $\mathbb{R}'$. But it’s easy to check that this funny field $\mathbb{C}'$ is algebraically closed and of characteristic zero. So, it’s isomorphic to $\mathbb{C}$.

In short, different ‘versions’ of the real numbers can give rise to the same version of the complex numbers! This is stuff they didn’t teach me in school.

All this is just background.

To a first approximation, Zilber considers uncountably categorical theories ‘logically perfect’. Let me paraphrase him:

There are purely mathematical arguments towards accepting the above for a definition of perfection. First, we note that the theory of the field of complex numbers (in fact any algebraically closed field) is uncountably categorical. So, the field of complex numbers is a perfect structure, and so are all objects of complex algebraic geometry by virtue of being definable in the field.

It is also remarkable that Morley’s theory of categoricity (and its extensions) exhibits strong regularities in models of categorical theories generally. First, the models have to be highly homogeneous, in a sense technically different from the one discussed for manifolds, but similar in spirit. Moreover, a notion of

dimension(the Morley rank) is applicable to definable subsets in uncountably categorical structures, which gives one a strong sense of working with curves, surfaces and so on in this very abstract setting. A theorem of the present author states more precisely that an uncountably categorical structure $M$ is either reducible to a 2-dimensional “pseudo-plane” with at least a 2-dimensional family of curves on it (so is non-linear), or is reducible to a linear structure like an (infinite dimensional) vector space, or to a simpler structure like a $G$-set for a discrete group $G$. This led to a Trichotomy Conjecture, which specifies that the non-linear case is reducible to algebraically closed fields, effectively implying that $M$ in this case is an object of algebraic geometry over an algebraically closed field.

I don’t understand this, but I believe that in rough terms this would amount to getting ahold of algebraic geometry from purely ‘logical’ principles, not starting from ideas in algebra or geometry!

Ehud Hrushovski showed that the Trichotomy Conjecture is false. However, Zilber has bounced back with a new improved notion of logically perfect theory, namely a ‘Noetherian Zariski theory’. This sounds like something out of algebraic geometry, but it’s really a concept from logic that *takes advantage of the eerie parallels between structures defined by uncountably categorical theories and algebraic geometry*.

Models of Noetherian Zariski theories include not only structures from algebraic geometry, but also from noncommutative algebraic geometry, like quantum tori. So, Zilber is now trying to investigate the foundations of physics using ideas from model theory. It seems like a long hard project that’s just getting started.

Here’s a concrete conjecture that illustrates how people are hoping algebraic geometry will spring forth from purely logical principles:

**The Algebraicity Conjecture.** Suppose $G$ is a simple group whose theory (consisting of all sentences in first-order theory of groups that hold for this group) is uncountably categorical. Then $G = \mathbb{G}(K)$ for some simple algebraic group $\mathbb{G}$ and some algebraically closed field $K$.

Zilber has a book on these ideas:

But there are many prerequisites I’m missing, and Richard Elwes, who studied with Zilber, has offered me some useful pointers:

If you want to really understand the Geometric Stability Theory referred to in your last two paragraphs, there’s a good (but hard!) book by that name by Anand Pillay. But you don’t need to go anywhere near that far to get a good idea of Morley’s Theorem and why the complex numbers are uncountably categorical. These notes look reasonable:

- Nick Ramsey, Morley’s categoricity theorem.
Basically the idea is that a theory is uncountably categorical if and only if two things hold: firstly there is a sensible notion of dimension (Morley rank) which can be assigned to every formula quantifying its complexity. In the example of the complex numbers Morley rank comes out to be pretty much the same thing as Zariski dimension. Secondly, there are no ‘Vaughtian pairs’ meaning, roughly, two bits of the structure whose size can vary independently. (Example: take the structure consisting of two disjoint non-interacting copies of the complex numbers. This is not uncountably categorical because you could set the two cardinalities independently.)

It is not too hard to see that the complex numbers have these two properties once you have the key fact of ‘quantifier elimination’, i.e. that any first order formula is equivalent to one with no quantifiers, meaning that all they can be are sets determined by the vanishing or non-vanishing of various polynomials. (Hence the connection to algebraic geometry.) In one dimension, basic facts about complex numbers tell us that every definable subset of $\mathbb{C}$ must therefore be either finite or co-finite. This is the definition of a

strongly minimal structure, which automatically implies both of the above properties without too much difficulty. So the complex numbers are not merely ‘perfect’ (though I’ve not heard this term before) but are the very best type of structure even among the uncountably categorical.

If you know anything else that could help me out, I’d love to hear it!

## Re: Uncountably Categorical Theories

So for any theory $T$, we have a function $g_T: \kappa \mapsto$ the number of realizations of $T$ of size $\kappa$, where $\kappa$ varies over cardinalities. I wonder if it’s reasonable to consider theories where $g_T$ grows slowly, rather than actually being bounded by $1$. (I don’t know how slowly, but $g_T(\kappa) \leq \kappa$ would be a start.)