Monster Model Theory

Question

I read somewhere (in texts of John Baldwin, e.g.), that every first-order theory $T$ has a "monster model," say M.

(1) Any small model $M$ of $T$ can be regarded as a submodel of M. (2) If $A$ and $B$ are small enough subsets of M, and there is a partial elementary map between $A$ and $B$, then this extends to an automorphism of M.

My questions: is (a) correct as I write it here ?; (b) how should I see $A$ and $B$ ? It is hard for me to believe that they could be just any subsets of M.

I would be very interested if someone explained it through, e.g., the theory $T$ of finite projective planes.

Answer 1

Yes, the monster model can be difficult to visualize at first, and indeed without certain set-theoretic assumptions it's not actually guaranteed to exist! This turns out not to be a problem in practice, and the role of the monster model is essentially linguistic (just like how set theorists talk about forcing extensions of the universe, even though that patently doesn't make sense), but is a detail which can often be missed at first. This note of Baldwin says a bit about this.

All a monster model really is is a saturated model of sufficiently large cardinality. Saturated models are homogeneous, which immediately implies the at-first-nonintuitive automorphism fact you mention; to come to grips with this, it might be a good idea to look first at other situations where homogeneity crops up, such as Fraisse limits. Note that the existence of a partial elementary map from $A$ to $B$ implies that $A$ and $B$, viewed as structures on their own (that is, forgetting the rest of $M$), are isomorphic; the point is that any two parts of $M$ which look like each other "locally" in fact look like each other "globally."

It's possible that the confusion here is stemming from the use of "partial elementary map" here. In this context, a partial elementary map from $A$ to $B$ is a partial function $f:M\rightarrow M$ such that:

• $dom(f)\supseteq A$, $ran(f)\supseteq B$ (in particular, $f$ is total on $A$!), and

• For each formula $\varphi$ and tuple $\overline{a}$ from $A$, we have $M\models\varphi(\overline{a})$ iff $M\models\varphi(f(\overline{a}))$.

This immediately implies (assuming the language is relational, so all subsets are substructures) that $f\upharpoonright A$ is an isomorphism from the substructure $A$ to the substructure $B$. (In the presence of function symbols, we have the slightly messier statement that $f\upharpoonright\hat{A}$ is an isomorphism from the substructure $\hat{A}$ to the substructure $\hat{B}$, where $\hat{C}$ is the smallest substructure of $M$ containing the subset $C$ - that is, the closure of $C$ under the functions of $M$.)

Answer 2

Let me try to address your question (elaborated in the comments to your question and to Noah's answer) about "monster projective planes".

You are interested in the theory $T$ of finite projective planes, by which I assume you mean the set of all sentences in the language of incidence structures $\{P,L,I\}$ which are true in all finite projective planes.

Here are some comments about the notion of a monster model for $T$:

1. First, let's note that we don't actually know what the theory $T$ is. For example, we don't even know whether there are any projective planes of order $12$. I certainly don't know how to axiomatize it.
2. The facts in your question about monster models are right, except that they should be stated for a complete first-order theory $T$ with infinite models. If $T$ is incomplete, it has multiple completions $T_1$ and $T_2$. If $\mathbf{M}$ is a monster model for $T_1$ and $M\models T_2$, then $\mathbf{M}$ and $M$ are not elementarily equivalent, so it's impossible to find an elementary embedding of $M$ into $\mathbf{M}$ (i.e. regard $M$ as an elementary submodel of $\mathbf{M}$).
3. Our theory $T$ is certainly not complete, because it has finite models of different sizes. Since there are arbitrarily large finite projective planes, $T$ also has infinite models by compactness.
4. Picking a monster model $\mathbf{M}\models T$ determines a completion $T' = \text{Th}(\mathbf{M})$ of $T$, and we should really think of $\mathbf{M}$ as a monster model for $T'$. Note that $T'$ no longer has finite models. Since we don't really understand $T$, it seems nontrivial to me to understand what the possible completions $T'$ of $T$ look like.
5. Certainly $\mathbf{M}$ is a huge projective plane. It contains as an elementary subplane all of the models of $T'$. Its substructures are just all the models of the universal consequences of $T'$. But again, to understand which projective planes appear as subplanes of $\mathbf{M}$, we have to understand the theory $T'$ (in particular, its universal consequences), which seems nontrivial.
6. As for homogeneity, we know that small subsets $A$ and $B$ are conjugate by an automorphism of $\mathbf{M}$ if and only if there is a partial elementary map $f\colon A\to B$. When does such an $f$ exist? The isomorphism type of the subplanes generated by $A$ and $B$ are encoded in the complete first-order types of $A$ and $B$, so it is certainly a necessary condition that $A$ and $B$ generate isomorphic subplanes. But this will typically not be sufficient. To fully answer the question (which is equivalent to the question "is $\text{tp}(A/\emptyset) = \text{tp}(B/\emptyset)$?") we need to understand what first-order formulas can express in models of $T'$, and this question is hard to answer without having a good understanding of $T'$.
7. Maybe you're actually interested in the theory of all projective planes, rather than the theory of finite projective planes described above. This theory has the advantage that it's easily axiomatizable, and it has some nice completions. If you just take an arbitrary completion $T'$ (i.e. pick an arbitrary monster model $\mathbf{M}$), the comments in points 5. and 6. still apply, since the theory of projective planes has lots of completions, some of which are going to be very hard to understand. For example, the theory of $\mathbb{P}^2(\mathbb{Q})$ is bi-interpretable with the theory of the rational field, which is as complicated as it gets from the point of view of model theory.
8. But among all the completions of the theory of projective planes, there are some gems. The nicest are (a) the theory of projective planes over algebraically closed fields of a fixed characteristic, and (b) the model companion of the theory of projective planes, i.e. the theory of existentially closed projective planes. The latter is the subject of a paper I wrote with Gabe Conant, and it's probably the one you want to look at if you're interested in the notion of a "monster projective plane" as a universal domain for the class of all projective planes. A monster model for the theory (b) really does have a copy of every small projective plane as a subplane. And in both theories (a) and (b), a necessary and sufficient condition for the existence of a partial elementary map $f\colon A\to B$ is that $A$ and $B$ generate isomorphic subplanes (so any two isomorphic copies of a projective plane $P$ in $\mathbf{M}$ are conjugate by an automorphism of $\mathbf{M}$). This fact comes down to the ability to understand first-order formulas and complete types by proving quantifier elimination "down to existential quantifiers over the subplane generated by the variables".