The n-Category Café

@Allen, there’s a lot of deep work around that sort of question which goes by the name of “classification theory”, and my understanding is that the various possibilities are now well understood.

The key result was a theorem of Shelah that there is a dichotomy, in that either there are the maximum possible number of models 2^k in each uncountable cardinality k, or there are “few”. Subsequently others have analysed exactly what “few” means. There are more details on this Wiki page.

One of the weirdest facts on that Wikipedia page was proved by Robert Vaught: the number of countably infinite models of a theory can be any nonnegative integer except 2.

(Of course, it can also be infinite.)

Thanks!

Are there ways to construct nonstandard models of the axioms for a real closed field that don’t use nonprincipal ultrafilters, and thus the axiom of choice (or at least some weakened version thereof)?

My mind boggles at nonprincipal ultrafilters: it seems at some intuitive level I refuse to think they exist. It’s a bit odd, because in some other contexts the axiom of choice seems perfectly fine to me. But nonprincipal ultafilters remind me too much of a thing that’s somewhere… but nowhere in particular. Sort of like that “damned elusive pimpernel” in The Scarlet Pimpernel:

They seek him here, they seek him there.

Those Frenchies seek him everywhere.

Is he in heaven or is he in hell?

That damned elusive pimpernel.

So I think it’s wrong to say that C is “unique” in any reasonable way.

I can’t agree with this statement! Most mathematicians would, I think, interpret “unique” as meaning “unique up to isomorphism”. Perhaps category theorists are atypical in this regard, I don’t know.

It’s perfectly true, as you say, properties of this type are essentially orthogonal to the property of “having lots of automorphisms”. (In fact one might even strengthen this to say the two are negatively correlated: the more automorphisms X has, the easier it may be for Y to be isomomorphic to X.)

Theo wrote:

No category theorist should for a moment consider “uncountable categoricity” to be any more “perfect” than the usual situation. Sure, every algebraically closed characteristic $0$ field of cardinality $\mathfrak{c}$ is isomorphic to $\mathbb{C}$. But not in any canonical way. (This is due, more or less, to the hugeness of the Galois group of $\mathbb{C}$ over $\mathbb{Q}$.) So I think it’s wrong to say that $\mathbb{C}$ is “unique” in any reasonable way.

Personally, instead of using morally loaded words like “perfect” or “reasonable” here, I prefer to just state the facts: all algebraically closed characteristic $0$ field of a fixed uncountable cardinality are isomorphic, but not in a unique way. So, the groupoid of these fields is connected, but not contractible. Thinking of it as a homotopy type, its $\pi_0$ is trivial, but not its $\pi_1$.

(I mentioned “perfection” not because I think that way, but because I was trying to explain Zilber’s thought.)

An orthogonal complaint is that the uncategorical categoricity of $\mathbb{C}$ is not going to explain, without much further elaboration, the role of $\mathbb{C}$ in quantum mechanics…

I agree with that. Physicists use the standard topology on $\mathbb{C}$ all over the place (like in electrical engineering), and quantum mechanics seems to rely fundamentally on its $\ast$-algebra structure. So, when it comes to physics, I think of Zilber’s work as a tentative first exploration of something that would require fundamentally new ideas to succeed. There’s certainly no obvious reason why being uncountably categorical is either a necessary or sufficient reason for a mathematical structure to be important in physics.

It might turn out that model theory has nothing interesting to say about mathematical physics. After all, in physics we’re often more interested in syntax, i.e., calculations starting from axioms. Since when do physicists really care about set-theoretic models of the axioms? For example: who among them really cares if the real numbers include infinitesimals or not? Newton used them; nothing bad happened when people stopped using them; nothing bad would happen if we started again. It doesn’t seem to affect any experimental predictions.

Or, it could be that model theory has interesting things to say about mathematical physics, but we just haven’t figured out what. This is what I’m actually hoping. It could easily take a century to get to the bottom of this. Maybe uncountable categoricity will turn out to be less important than some other logical properties of theories.

When it comes to algebraic geometry rather than physics, the model theorists seem to be on much firmer ground. Hrushovski has already used model theoretic techniques to prove the geometric Mordell–Lang conjecture in all characteristics. Algebraic geometry more often avoids working with the usual topology and $\ast$-algebra structure on $\mathbb{C}$. So maybe the slogan “uncountably categorical $\approx$ logically perfect” is suited to algebraic geometry, not physics.

Good question! I have no idea. Regarding $\mathbb{C}$, the key result seems to be Steinitz’s Theorem saying that the isomorphism type of an algebraically closed field is determined by its characteristic and transcendence degree… plus the fact that for uncountable algebraically closed fields the transcendence degree is the same as the cardinality. I don’t know if people have analyzed the use of choice in these results.

Somewhat separately, one could ask if Morley’s categoricity theorem relies on the axiom of choice.

I suspect that whenever “transcendence degree” enters the picture, so will AC, since transcendence degree is the dimension of a vector space, and “every vector space has a basis” is equivalent to AC.

Yes, that’s indeed a “kick in the teeth”. I mentioned it in my post:

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.

Put differently, the complex numbers contain more than one non-isomorphic real-closed field that together with $i$ generate all of it.

I can write out this (short!) proof if you want; I think you’ll like it, if you don’t know it already.

Please do!

I’d be happy to hear that proof. I somehow managed to avoid learning a lot of commutative algebra, so I can’t help you with you question.

Does ‘morally trivial’ mean ‘trivial modulo uninteresting technical details’ (in analogy to ‘morally true’), or does it mean ‘having little deep significance’ (that is, ‘moral content’)?

John, I guess your first interpretation of Martin’s “morally trivial” is closer to what he intended. I took it to mean that the Hilbert basis theorem is deep, but this set-theoretic fact about fields isn’t (though myself, I don’t even know a proof). I should just ask him.

Here goes with the logician’s Nullstellensatz. By the looks of my notes, Martin did a lot of it out loud rather than on the board, so there is a distinct possibility of mistakes.

We’ll prove: for any proper ideal $I$ of $\mathbb{C}[x_1, \ldots, x_n]$, the set $$V(I) = \{ a \in \mathbb{C}^n : f(a) = 0 \,\,\text{ for all }\,\, f \in I\}$$ has at least one point.

1. Hilbert’s basis theorem implies that any such $I$ is finitely generated — say by $f_1, \ldots, f_k$. We want to show that this statement $\phi$: $$\exists x_1, \ldots, x_n. f_1(x_1, \ldots, x_n) = \cdots = f_k(x_1, \ldots, x_n) = 0$$ holds in $\mathbb{C}$.

2. Take a maximal ideal $m$ containing $I$. Then $\phi$ holds in the field $R = \mathbb{C}[x_1, \ldots, x_n]/m$, and therefore also holds in the algebraic closure $\bar{R}$ of $R$ (which is an extension of $\mathbb{C}$).

   (So, we want to show that if some existential statement $\phi$ holds in an extension of $\mathbb{C}$, then it holds in $\mathbb{C}$ itself. That is, we want $\mathbb{C}$ to be “existentially closed” — “algebraically closed $\implies$ existentially closed”.)

3. [Here my notes look a bit unreliable, but I think the point is as follows.] Now consider ideals generated by a fixed number $k$ of polynomials of a fixed maximal degree $d$, and let $\psi$ be the proposition: $$\begin{aligned} \text{for all polynomials }\,\, f_1, \ldots, f_k \,\,\text{ of degree at most }\,\, d. \exists \mathbf{x}.\\ (f_1, \ldots, f_k) \,\,\text{ is a proper ideal } \implies f_1(\mathbf{x}) = \cdots = f_k(\mathbf{x}) = 0. \end{aligned}$$ (Martin remarked that the part of that proposition strictly after “$\exists \mathbf{x}$” is “actually quantifier-free”, but I don’t see why that’s true and it doesn’t seem to be needed in what follows.)

4. Set-theoretic thought: any two algebraically closed fields of a given characteristic and of the same uncountable cardinality are isomorphic.

5. But $\psi$ is true in one algebraically closed field of continuum cardinality and characteristic zero (namely, $\bar{R}$), so it’s true in $\mathbb{C}$ too.

   Ta-da! (Martin didn’t say that.)

With luck a logician can give you more details. But one intuition is that $\mathbb{C}$ doesn’t “know” whether a number is or isn’t in $\mathbb{N}$. The language is different.

Here’s a manifestation: you say “$\mathbb{N}$ is the additive closure of $\{0,1\}$”, which I will rewrite as “a number $c\in \mathbb{C}$ is in $\mathbb{N}$ if it is a sum of some number of $1$s”. But this isn’t a 1st-order sentence. How many $1$s? A natural number of them … but that’s what we’re trying to define. Actually, you were more precise: you said “the smallest closure …”. But to define that, you need to quantify over subsets of $\mathbb{C}$ with some property, and that’s the very essence of second-order mathematics.

Bruce wrote:

But if you did, then this quality of “perfection” is not so much about $\mathbb{C}$ vs. $\mathbb{N}$ as about which set of axioms you feel comfortable using at the time.

Right.

For $\mathbb{N}$, I said that the Peano axioms are incomplete; it’s also undecidable if a statement follows from those axioms.

For $\mathbb{C}$, I said that the axioms for an algebraically closed field of characteristic zero are complete; they’re also decidable. These axioms are merely:

• the ‘field’ axioms: the usual algebraic identities involving $+, \times, 0$ and $1$ together with laws saying that anything has an additive inverse and anything except 0 has a multiplicative inverse.

• the ‘algebraically closed’ axiom, saying any polynomial has a root.

• the ‘characteristic zero’ axiom schema, saying that $1 \ne 0$, $1 + 1\ne 0$, $1 + 1 + 1 \ne 0$, etcetera.

Note we have nothing like mathematical induction or the concept of ‘subset of complex numbers’. We can’t define exponentiation, so we can’t define $\pi$, much less prove that it exists.

Indeed, if we restrict attention to algebraic complex numbers (roots of polynomials with integer coefficient), we get a perfectly fine countable model of this theory. We’d get another, nonisomorphic countable model if we took the smallest algebraically closed field containing the algebraic complex numbers together with one transcendental number, like $\pi$. Or, we could throw in both $\pi$ and $e$, which are both transcendental and ‘independent’: they don’t satisfy any nontrivial polynomial relations with each other.

So, there are tons of different countable models. This makes Steinitz’s Theorem seem rather wonderful: up to isomorphism, there’s just one model of each uncountable cardinality.

But in fact, maybe it’s not so wonderful. I believe all these models are just the algebraic complex numbers with some uncountable number of independent transcendentals thrown in.

What makes the complex numbers more interesting than ‘algebraic numbers with an enormous wad of independent extra numbers thrown in’ is their topology… or equivalently, complex conjugation (which allows us to define the distance function $|x - y|$).

Zilber’s approach to treating the complex numbers as merely an algebraically closed field of characteristic zero is quite common in algebraic geometry (which actually spends a lot of time exploring the relation between this attitude and the attitude where you also study the topology).

You might think the incompleteness and undecidability of the natural numbers are due to including the principle of mathematical induction in the Peano axioms. But Robinson arithmetic leaves out induction and is still incomplete and undecidable! So it’s a bit subtle.

Perhaps related — is it ruled out that any “perfect” theory can include topology, reals, etc, or merely not yet known whether this is possible?

There’s a lot known about this kind of stuff, but I don’t know most of it. With his Trichotomy Conjecture and subsequent refinements, Zilber and others are working toward a kind of ‘classification’ of uncountably categorical structures. Very very roughly, the Trichotomy Structure and its corrected versions say all ‘logically perfect’ theories involve either 1) a linear order, 2) a vector space over a division ring, or 3) an algebraically closed field.

If we want to get ordinary physics into the game, I think we need something like the real numbers seen not just as a real closed field but as a complete ordered field. There are some standard axioms for an ordered field. If we use second-order logic (which is rather unpopular) we can quantify over predicates and add an extra axiom saying “for all predicates $P$, if for some $x$ we have $P(y)$ only for $y \le x$, then there is a least $x$ such that $P(y)$ holds only for $y \le x$.”

The resulting theory is categorical: it has only one model! So, in some sense this is even more “perfect” than the uncountably categorical theories we’ve been discussing. But many logicians dislike second-order logic because there’s not such a nice match of syntax and semantics as in first-order logic. Unlike first-order logic, second-order logic has no system of deduction rules that’s:

• Sound: Every provable second-order sentence holds in every model (using the standard concept of model).

• Complete: Every formula that holds in every model is provable.

• Effective: There is an algorithm that can correctly decide whether any given sequence of symbols is a valid proof.

So, most logicians consider second-order logic too defective to be happy that — unlike first-order logic — it admits theories that have a unique (up to isomorphism) infinite model.