# 6.4 Stability Theory Overview

Stability theory studies the complexity of first order theories by analyzing how types behave over parameter sets, and in particular by measuring how many distinct types exist and how these types interact with definable sets. The guiding idea is that a theory should be considered tame if the space of types is controlled, and wild if the space of types grows too quickly or exhibits arbitrary combinatorial patterns.

Throughout this section, let $T$ be a complete first order theory in a language $L$, and let $\mathcal M \models T$ be a sufficiently saturated model so that types over small parameter sets can be realized and studied internally.

### Counting Types

Let $A \subseteq M$ be a parameter set. The set:
$$
S_1(A)
$$
denotes the collection of all complete $1$-types over $A$ that are consistent with $T$.

Each element of $S_1(A)$ represents a possible way that a single element can behave with respect to all definable subsets over $A$. Thus the size of $S_1(A)$ measures how many distinct behaviors are possible.

If $|S_1(A)|$ is small compared to $|A|$, then the theory exhibits a form of uniformity. If $|S_1(A)|$ is very large, then the theory allows many different behaviors and is correspondingly complex.

### Definition 6.61 (Stable Theory)

The theory $T$ is stable if for every infinite set $A$:
$$
|S_1(A)| \leq |A|.
$$

This condition ensures that the number of possible types over a set does not grow faster than the size of the set itself, and it is the basic criterion for tameness in model theory.

If there exists some infinite $A$ such that:
$$
|S_1(A)| > |A|,
$$
then $T$ is unstable.

### Interpretation of Stability

Stability can be understood as a restriction on how formulas can distinguish between elements. In a stable theory, formulas cannot encode arbitrary patterns over large parameter sets, and therefore the number of distinct types remains controlled.

This restriction leads to a setting where types can often be described in a uniform and definable way, which allows one to develop a structural theory.

### Order Property

The main combinatorial source of instability is the order property.

### Definition 6.62 (Order Property)

A formula $\varphi(x,y)$ has the order property if there exist sequences:
$$
(a_i : i \in \mathbb N), \quad (b_j : j \in \mathbb N)
$$
in some model of $T$ such that:
$$
\mathcal M \models \varphi(a_i,b_j)
\quad \text{if and only if} \quad
i < j.
$$

This configuration encodes an infinite linear order inside the behavior of the formula, allowing the formula to distinguish infinitely many different configurations.

### Lemma 6.63 (Order Property Implies Instability)

If some formula in $T$ has the order property, then $T$ is unstable.

Proof

Suppose $\varphi(x,y)$ has the order property, witnessed by sequences $(a_i)$ and $(b_j)$.

Let:
$$
A = \{b_j : j \in \mathbb N\}.
$$

For each subset $I \subseteq \mathbb N$, define a partial type:
$$
p_I(x) = \{\varphi(x,b_j) : j \in I\} \cup \{¬\varphi(x,b_j) : j \notin I\}.
$$

We claim that each $p_I(x)$ is finitely satisfiable. Indeed, given finitely many conditions, only finitely many indices $j$ are involved, and one can choose an index $i$ large enough so that the pattern of truth values $\varphi(a_i,b_j)$ matches the required conditions, using the defining property:
$$
\mathcal M \models \varphi(a_i,b_j)
\quad \text{if and only if} \quad
i < j.
$$

Thus every finite subset of $p_I(x)$ is realized by some $a_i$, so $p_I(x)$ is finitely satisfiable.

By compactness, each $p_I(x)$ extends to a complete type over $A$, and distinct subsets $I$ give rise to distinct types.

Since there are:
$$
2^{\aleph_0}
$$
subsets of $\mathbb N$, we obtain:
$$
|S_1(A)| \geq 2^{\aleph_0}.
$$

Because $A$ is countable, this shows:
$$
|S_1(A)| > |A|,
$$
and therefore $T$ is unstable.

### Stability Excludes the Order Property

The previous lemma shows that the order property is incompatible with stability. In fact, one can show that stability is equivalent to the absence of the order property, although the full equivalence requires additional work.

### Example 6.64

The theory of dense linear orders without endpoints is unstable.

Indeed, for a dense order, one can construct sequences that witness the order property using the ordering relation itself, and therefore the theory admits many distinct types corresponding to cuts in the order.

### Example 6.65

The theory of algebraically closed fields is stable.

In this theory, types over a parameter set are controlled by algebraic conditions, and the number of possible types is bounded by the size of the parameter set.

This reflects the algebraic nature of the theory, where elements are constrained by polynomial equations.

### Definability of Types

One of the most important structural properties of stable theories is that types over models are definable.

### Definition 6.66 (Definable Type)

Let $A \subseteq M$, and let $p(x) \in S_1(A)$. The type $p(x)$ is definable if for every formula $\varphi(x,y)$ there exists a formula:
$$
d_\varphi(y)
$$
with parameters from $A$ such that for every $b \in A$:
$$
\varphi(x,b) \in p(x)
\quad \text{if and only if} \quad
\mathcal M \models d_\varphi(b).
$$

Thus membership of instances of $\varphi$ in the type is determined by a definable condition on the parameter.

### Lemma 6.67 (Definability of Types in Stable Theories)

If $T$ is stable and $\mathcal M \models T$, then every complete type over $M$ is definable.

Proof

The proof uses the stability assumption to control the number of possible behaviors of formulas. For each formula $\varphi(x,y)$, consider the set of parameters $b$ such that $\varphi(x,b)$ belongs to the type. Stability ensures that this set can be described by a formula, because otherwise one could construct too many distinct types by varying the parameter.

More precisely, if definability failed, then one could encode many distinct patterns of inclusion and exclusion of formulas, leading to a large number of distinct types over $M$, which would contradict the stability condition:
$$
|S_1(M)| \leq |M|.
$$

Thus definability of types follows from the boundedness of type spaces.

### Independence and Forking

Stability allows one to define a notion of independence between elements and parameter sets, called forking independence, which generalizes algebraic independence.

Given $A \subseteq B \subseteq M$ and a tuple $a$, one says that the type of $a$ over $B$ does not fork over $A$ if it does not introduce additional complexity beyond what is already determined over $A$.

Forking independence satisfies several important properties in stable theories:

symmetry, meaning that independence is mutual,

transitivity, meaning that independence can be composed,

extension, meaning that types can be extended without introducing dependence.

These properties allow one to treat independence as a geometric notion.

### Morley Rank

In stable theories, definable sets can be assigned a notion of dimension called Morley rank, which measures their complexity in a way analogous to dimension in geometry.

Morley rank is defined inductively by considering how a definable set can be partitioned into smaller definable subsets, and it takes values in the ordinals.

This notion provides a powerful invariant for analyzing definable sets and plays a central role in geometric stability theory.

### Strongly Minimal Sets

A definable set is strongly minimal if every definable subset is either finite or cofinite.

Such sets are the simplest infinite definable sets, and they behave like one dimensional geometric objects.

In a strongly minimal structure, every element is either algebraic over a given set or independent from it, and this leads to a very rigid structure.

### Lemma 6.68 (Bound on Types)

If $T$ is stable and $A$ is infinite, then for every natural number $n$:
$$
|S_n(A)| \leq |A|.
$$

Proof

The proof proceeds by induction on $n$. For $n=1$, this is the definition of stability.

Assume the result holds for $n$. Consider types in $n+1$ variables. Such a type can be viewed as extending a type in $n$ variables by specifying the behavior of an additional coordinate.

For each choice of the first $n$ coordinates, there are at most $|A|$ possibilities for the remaining coordinate, by stability.

Thus the total number of $(n+1)$-types is bounded by:
$$
|A| \cdot |A| = |A|.
$$

Hence the bound holds for all finite $n$.

### Consequences of Stability

Stability leads to a rich structural theory in which models can often be analyzed using geometric intuition.

Types correspond to points, definable sets correspond to geometric objects, and independence corresponds to geometric independence.

This framework allows one to apply methods from algebra and geometry to the study of logical structures.

### Perspective

Stability theory provides the first major step in the classification of theories, and it identifies a class of theories where a detailed and robust structural analysis is possible. It also introduces many of the tools, such as types, independence, and dimension, that are used throughout modern model theory.