Let $T$ be strongly minimal and let $M\models T$. For every $A\subseteq M$, every set $I\subseteq M$ that is independent over $A$ extends to a basis of $M$ over $A$, and any two bases of $M$ over $A$ have the same cardinality.

For a complete theory $T$, the following are equivalent: 1. $T$ is stable in some infinite cardinal $\kappa$ with $\kappa \ge |T|$. 2. For every infinite cardinal $\kappa$ satisfying $\kappa^{|T|} = \kappa$, the theory $T$ is stable in $\kappa$. 3. No formula has the order property with respect to $T$.

Let $T$ be a complete stable first-order theory, let $\mathfrak C \models T$ be a sufficiently saturated and strongly homogeneous monster model, let $M \preccurlyeq \mathfrak C$ be a model of $T$, and let $p(x) \in S_x(M)$ be a complete type over $M$. Then $p$ is definable over $M$: for every formula $\varphi(x;y)$ there exists an $M$-formula $d_p\varphi(y)$ such that, for every $b \in M^{|y|}$, \begin{align*} M \models d_p\varphi(b) \quad \Longleftrightarrow \quad \varphi(x;b) \in p. \end{align*}

Let $M \models T$ be the ambient structure containing $A$, and let $T_A := \operatorname{Th}_{\mathcal{L}(A)}((M,a)_{a \in A})$. If $\mathcal{B}_n(A)$ denotes the Boolean algebra of $\mathcal{L}(A)$-formulas in variables $x_1,\dots,x_n$ modulo the relation $\varphi \equiv_{T,A} \psi$ iff $T_A \models \forall x_1 \cdots \forall x_n(\varphi \leftrightarrow \psi)$, then the map \begin{align*} \mathcal{B}_n(A) &\longrightarrow \operatorname{Clop}(S_n(A)),\\ [\varphi]_{\equiv_{T,A}} &\longmapsto [\varphi] \end{align*} is a Boolean algebra isomorphism from $\mathcal{B}_n(A)$ onto the Boolean algebra of clopen subsets of $S_n(A)$.

Let $T$ be a complete theory in a countable language $\mathcal{L}$. For each integer $n \geq 1$, let $S_n(T)$ denote the Stone space of complete $n$-types over $T$, with basic open sets of the form \begin{align*} [\varphi] := \{p \in S_n(T) : \varphi \in p\} \end{align*} for $\mathcal{L}$-formulas $\varphi(x_1,\dots,x_n)$. Then $T$ has a countable atomic model if and only if, for every integer $n \geq 1$, the isolated types are dense in $S_n(T)$.

Let $T$ be a complete countable first-order theory in a countable language $L$, and suppose that $T$ is totally transcendental. Let $\mathfrak{C} \models T$ be a sufficiently saturated monster model, and let $A \subseteq \mathfrak{C}$ be countable. Let $L(A)$ be the expansion of $L$ by a constant symbol $c_a$ for each $a \in A$, and let $T_A$ be the complete $L(A)$-theory of $\mathfrak{C}$ with $c_a$ interpreted as $a$. Then $T_A$ has a prime model. Equivalently, there exists an elementary substructure $M \preccurlyeq \mathfrak{C}$ with $A \subseteq M$ such that for every model $N \models T$ containing $A$, there is an elementary embedding $f: M \to N$ satisfying $f(a) = a$ for every $a \in A$. Moreover, any two models prime over $A$ are isomorphic over $A$.

Let $L$ be a first-order language, let $T$ be an $L$-theory, and let $A$ be a parameter set, viewed in the expanded language $L(A)$ by adding a constant symbol for each element of $A$. Let $x = (x_1,\dots,x_n)$ be a tuple of variables, and let $\Sigma(x)$ be a set of $L(A)$-formulas with free variables among $x_1,\dots,x_n$. Assume that $\Sigma(x)$ is finitely satisfiable over $T \cup \operatorname{Diag}(A)$, meaning that for every finite subset $\Sigma_0(x) \subset \Sigma(x)$, the theory \begin{align*} T \cup \operatorname{Diag}(A) \cup \{\exists x\, \bigwedge_{\varphi \in \Sigma_0} \varphi(x)\} \end{align*} is satisfiable. Then there exists an $L(A)$-structure $N$ such that $N \models T \cup \operatorname{Diag}(A)$ and a tuple $b = (b_1,\dots,b_n) \in N^n$ such that \begin{align*} N \models \varphi(b) \end{align*}…

Let $L$ be a first-order language, let $M$ be an $L$-structure, let $A \subseteq M$ be a parameter set, and let $n \in \mathbb{N}$. Let $S_n(A)$ denote the Stone space of complete $n$-types over $A$ in variables $\bar{x} = (x_1,\dots,x_n)$, equipped with the Stone topology whose basic open sets are \begin{align*} [\theta] := \{p \in S_n(A) : \theta(\bar{x}) \in p\}, \end{align*} where $\theta(\bar{x})$ ranges over all $L(A)$-formulas in the variables $\bar{x}$. For every $L(A)$-formula $\varphi(\bar{x})$, the subset $[\varphi] \subseteq S_n(A)$ is both open and closed. Moreover, for all $L(A)$-formulas $\varphi(\bar{x})$ and $\psi(\bar{x})$, \begin{align*} [\neg \varphi] &= S_n(A) \setminus [\varphi], \\ [\varphi \wedge \psi] &= [\varphi] \cap [\psi], \\ [\varphi \vee \psi] &= [\varphi] \cup [\psi]. \end{align*}…

Let $T$ be a complete first-order theory in a language $\mathcal{L}$. If $M \models T$ is countable and atomic, meaning that for every finite tuple $a \in M^n$ the complete type $\operatorname{tp}^M(a/\varnothing)$ is isolated by an $\mathcal{L}$-formula, then $M$ is a prime model of $T$: for every model $N \models T$, there exists an elementary embedding $f: M \to N$.

Let $D$ be a parameter-free definable strongly minimal set in a monster model $\mathfrak{M}$ of a complete theory. For $A \subset D$, define \begin{align*} \operatorname{cl}: \mathcal{P}(D) \to \mathcal{P}(D), \qquad \operatorname{cl}(A)=D \cap \operatorname{acl}_{\mathfrak{M}}(A). \end{align*} Then $\operatorname{cl}$ is a pregeometry on $D$: it is extensive, increasing, idempotent, has finite character, and satisfies exchange.