My previous answer addressed the question "How can I make sense of the counting types definition of stability outside of a first-order model theory context?"

After discussion in the comments, I realized that you really wanted to know what stability means in the context of an (abstract) projective plane. So I'll try to address that question in this answer.

First some background in model theory, just to make sure if we're on the same page. Let $M$ be an $L$-structure and $C$ a subset of $M$. We write $L(C)$ for the set of all first-order formulas with parameters from $C$. For example, if $L = \{P,L,I\}$ is the language of incidence structures (where $P$ and $L$ are unary relations picking out the points and lines and $I$ is the binary incidence relation), then $\varphi(x_1,x_2): \forall y\, ((L(y) \land I(x_1,y)\land I(x_2,y))\rightarrow \lnot I(c,y))$ is a formula with a single parameter $c\in C$ expressing that no line through $x_1$ and $x_2$ is incident with $c$.

Let $x = (x_1,\dots,x_n)$ be a tuple of variables. A *complete type* $p(x)$ over $C$ (in variable context $x$) is a maximal set of $L(C)$-formulas with free variables from $x$ which is consistent with $\mathrm{Th}_{L(C)}(M)$, the set of all $L(C)$-sentences true in $M$. Maximality amounts to saying that for every formula $\varphi(x)\in L(C)$, either $\varphi(x)\in p(x)$ or $\lnot \varphi(x)\in p(x)$. We write $S_x(C)$ for the set of all complete types over $C$ in context $x$.

Let $a = (a_1,\dots,a_n)$ be an element of $M$. Then the *type of $a$ over $C$* is the set of all formulas with parameters from $C$ satisfied by $a$ in $M$: $$\mathrm{tp}(a/C) = \{\varphi(x)\in L(C)\mid M\models \varphi(a)\}.$$

We could equivalently define $S_x(C) = \{\mathrm{tp}(b/C)\mid b\in N, M\preceq N\}$. Note here that in general we have to pass to an elementary extension of $M$: it may be that not every complete type over $C$ is the type of an element in $M$, but every complete type is realized in some elementary extension.

Now a theory $T$ is *stable* if there is some infinite cardinal $\kappa$ such that for all models $M\models T$, all sets $C\subseteq M$ with $|C| = \kappa$, and all variable contexts $x$, we have $|S_x(C)| = \kappa$.

In Shelah's slides, he gives a slightly different definition, depending on the following facts:

- It suffices to check stability when the context is a single variable $x$ instead of a tuple.
- It suffices to check stability when $C$ is an (elementary) submodel.
- If $T$ is stable, then actually we have that for all infinite $\kappa$, $|C| = \kappa$ implies $|S_x(C)| = \kappa^{|L|}$, where $|L|$ is the number of formulas in the language (with no parameters). The fact that there are cardinals $\kappa$ such that $\kappa^{|L|} = \kappa$ gives the converse. Usually we're most interested in the case $|L| = \aleph_0$.

In practice, if you want to prove that a theory is stable by counting types, then you need some way to understand all the possible complete types. This can be hard, because first-order formulas can be very complicated in general. There are essentially two ways of doing this:

*Quantifier elimination.* A theory $T$ has quantifier elimination if for every first-order formula $\varphi(x)$, there is a quantifier-free formula $\psi(x)$ such that $T\models \forall x\, \varphi(x)\leftrightarrow \psi(x)$. Since quantifier-free formulas are usually much easier to understand than general formulas (they are just Boolean combinations of the atomic formulas - basic relations and equalities between terms), quantifier elimination can be very helpful for counting types. Some theories admit weaker forms of quantifier elimination, where you can show that every formula is equivalent to a Boolean combination of formulas in some restricted class that are still easy to understand.

*Automorphisms.* If $a,a'\in M$ and $\sigma\colon M\to M$ is an automorphism with $\sigma(c) = c$ for all $c\in C$ and $\sigma(a) = a'$, then automatically $\mathrm{tp}(a/C) = \mathrm{tp}(a'/C)$. So upper bounds on the number of orbits of automorphism groups of models of $T$ gives upper bounds on the number of complete types.

OK, now let's look at the example of projective planes. The easiest example of a stable theory of projective planes is the complete theory $T$ of $\mathbb{P}^2(k)$, where $k$ is an algebraically closed field. One way to see that $T$ is stable is to show that it is bi-interpretable with the complete theory of $k$. Any theory of algebraically closed fields is stable, and stability is preserved under bi-interpretation.

$T$ does not have quantifier elimination, but if you add a binary function $f$ to the language such that $f(p_1,p_2)$ is the unique line through the points $p_1$ and $p_2$, and $f(l_1,l_2)$ is the unique point at the intersection of $l_1$ and $l_2$, then $T$ has quantifier elimination in the expanded language (this fact rests heavily on the fact that the theory of algebraically closed fields has quantifier elimination).

Let's think about the counting types criterion. Let $M\models T$ (then $M = \mathbb{P}^2(K)$ where $K$ is algebraically closed of the same characteristic as $k$). Suppose $|M| = \kappa$. How big is $S_x(M)$ (where $x$ is a single variable)?

A type over $M$ could be the type of a point or a line. It suffices to count the types of points, by duality. It turns out that if we coordinatize $M = \mathbb{P}^2(K)$, the type of a point over $M$ is completely determined by the algebraic relationships between the coordinates which are definable over $K$. In short, the types are exactly the scheme-theoretic points of $\mathbb{P}^2(K)$.

The points of dimension $0$ are the types of points in $M$ (there are $\kappa$-many of these). The points of dimension $1$ are the generic types of curves in the projective plane defined over $K$ (there are $\kappa$-many of these, since each can be associated with a homogeneous polynomial over $K$). And there is a unique point of dimension $2$, the generic type of the plane. Altogether, there are only $\kappa$-many types, so $T$ is stable (in fact, it is $\omega$-stable, meaning that $|M| = \kappa$ implies $|S_x(M)| = \kappa$ for *all* infinite cardinals $\kappa$).

When the field is not algebraically closed, we do not get such a nice description of types thanks to the loss of quantifier elimination in the field (though understanding types relative to the theory of $\mathbb{P}^2(k)$ can still be reduced to understanding types relative to the theory of $k$).

The situation is even worse for non-Desarguesian projective planes, which are not coordinatized by fields or even division rings. As I alluded to in the comments, however, Hrushovski constructions can be used to produce non-Desarguesian projective planes with stable theories. These structures are extremely homogeneous, which allows stability to be checked using the automorphism criterion I described above.

A Course in Model Theory. $\endgroup$ – Alex Kruckman May 29 '18 at 15:27