Whistle Stop Tour of Stability Theory - Model Theory

17 Whistle Stop Tour of Stability Theory

Definition 17.1. Given $κ \geq | L | + ℵ_{0}$ , we say $T$ is $κ$ -stable if for any $M ⊨ T$ , $| M | = κ$ we have $| S_{1} (M) | = κ$ .

We say $T$ is stable if it is $κ$ -stable for some $κ$ .

Example.

(1) ${ACF}_{p}$ , $TFDAG$ ( $ℚ$ -vector spaces) are $κ$ -stable for all $κ \geq ℵ_{0}$ .
(2) (Exercise) $T = T h (ℤ, +, 0, 1, {(\equiv_{n})}_{n \geq 2})$ (where $\equiv_{n}$ is congruence modulo $n$ ). This is $κ$ -stable for $κ > 2^{ℵ_{0}}$ .
(3) If $M ⊨ R G$ then $| S_{1} (M) | = 2^{| M |}$ .

Fact: $ℵ_{0}$ -stable theories have saturated models of all infinite cardinalities.

Definition 17.2. Let $φ (x, y)$ be an $L$ -formula, $x, y$ types of finite length.

We say $φ (x, y)$ has the order property with respect to $T$ if there is some $M ⊨ T$ , ${(a_{i})}_{i \geq 0}$ , ${(b_{j})}_{j \geq 0}$ such that $M ⊨ φ (a_{i}, b_{j})$ if and only if $i < j$ .

Example. $DLO$ has the order property, choose $(ℚ, <) = M$ and $a_{i} = b_{i} = i$ as your sequence.

Theorem 17.3 (Fundamental Theorem of Stability (light)). The following are equivalent:

(i)
$T$ is stable.
(ii)
No $L$ -formula has the order property with respect to $T$ .
(iii)
For any $M ⊨ T$ , every $p \in S_{n} (M)$ is definable.
(iv)
Non-forking is an independence relation.

Definition 17.4. A theory $T$ is strongly minimal if $\forall M ⊨ T$ every definable subset of $M$ is finite or cofinite.

Remark. $T$ strongly minimal implires $T$ is stable (count types).

Definition 17.5. Let $M ⊨ T$ , $A \subseteq M$ . Then $b \in acl (A)$ if there is an $L_{A}$ -formula $ϕ (x)$ such that

M ⊨ \exists_{x}^{= n} ϕ (x)

and $M ⊨ ϕ (b)$ .

Example 17.6. Let $T$ be strongly minimal. Then $acl$ has the exchange property:

a \in acl (B c) ∖ acl (B) ⟹ c \in acl (B a) .