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 $L$ 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 $M \models T$ 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 countable stable theory, let $A$ be a small parameter set, and let $D$ be an $A$-definable strongly minimal set. Assume that for every independent set $I \subset D$ there is a model $M_I$ prime and atomic over $A \cup I$, and that finite independent tuples from $D$ of the same length have the same type over $A$. If $I,J \subset D$ are independent sets with $|I|=|J|$, then $M_I \cong_A M_J$.

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 $T$ be a complete first-order theory in a countable language. Suppose there is a formula $\varphi(x;y)$ of $T$, where $x$ and $y$ are finite tuples of variables, with the order property: for every $n \in \mathbb{N}$ there exist tuples $a_1,\dots,a_n$ and $b_1,\dots,b_n$ in some model of $T$ such that \begin{align*} \models \varphi(a_i;b_j) \quad \Longleftrightarrow \quad i < j \end{align*} for all $1 \le i,j \le n$. Then for every uncountable cardinal $\kappa$, there exist two non-isomorphic models $M,N \models T$ with $|M| = |N| = \kappa$.

Let $L$ be a first-order language, let $A$ be a set of parameters for $L$, and let $n \in \mathbb{N}$. For each $L(A)$-formula $\varphi(\bar{x})$ in the tuple of variables $\bar{x} = (x_1,\dots,x_n)$, define \begin{align*} [\varphi] := \{p \in S_n(A) : \varphi(\bar{x}) \in p\}. \end{align*} Then the family \begin{align*} \mathcal{B} := \{[\varphi] : \varphi(\bar{x}) \text{ is an } L(A)\text{-formula}\} \end{align*} is a basis for the Stone topology on $S_n(A)$, and every member of $\mathcal{B}$ is clopen.

Let $L$ be a first-order language, let $T$ be an $L$-theory, let $x$ be a finite tuple of variables, and let $p(x)$ be a non-principal type over $T$. If $\theta(x)$ is an $L$-formula such that \begin{align*} T \cup \{\exists x\,\theta(x)\} \end{align*} is satisfiable, then there exists a formula $\varphi(x) \in p(x)$ such that \begin{align*} T \cup \{\exists x\,(\theta(x) \wedge \neg \varphi(x))\} \end{align*} is satisfiable.

Let $T$ be a complete stable first-order theory, and let $\mathfrak{C} \models T$ be a sufficiently saturated and sufficiently homogeneous monster model. For small parameter sets $A,B,C,D \subset \mathfrak{C}$, write $A \mathop{\perp}\limits_C B$ to mean that $\operatorname{tp}(A/BC)$ does not fork over $C$. Forking independence in $T$ satisfies the following properties: 1. Invariance: if $\sigma \in \operatorname{Aut}(\mathfrak{C})$ and $A \mathop{\perp}\limits_C B$, then $\sigma(A) \mathop{\perp}\limits_{\sigma(C)} \sigma(B)$. 2. Monotonicity: if $A \mathop{\perp}\limits_C BD$, then $A \mathop{\perp}\limits_C B$ and $A \mathop{\perp}\limits_{BC} D$. 3. Symmetry: if $A \mathop{\perp}\limits_C B$, then $B \mathop{\perp}\limits_C A$. 4. Transitivity: if $C \subset B \subset D$, then $A \mathop{\perp}\limits_C D$…

Let $T$ be a complete first-order theory in a language $\mathcal{L}$, let $M\models T$, let $A\subseteq M$ be a set of parameters, and let $x=(x_1,\dots,x_n)$ be a tuple of variables. Let $T_A:=\operatorname{Th}_{\mathcal{L}(A)}((M,a)_{a\in A})$ be the complete theory of the named-parameter structure. Let $\mathcal{B}_n(A)$ be the Boolean algebra of $\mathcal{L}(A)$-formulas in the free variables among $x$, modulo the [equivalence relation](/page/Equivalence%20Relation) \begin{align*} \varphi \equiv_{T,A} \psi \quad \Longleftrightarrow \quad T_A \models \forall x\,(\varphi(x) \leftrightarrow \psi(x)). \end{align*} For each complete $n$-type $p\in S_n(A)$, define \begin{align*} U_p:=\{[\varphi]_{\equiv_{T,A}}\in\mathcal{B}_n(A):\varphi\in p\}. \end{align*} Then $U_p$ is an ultrafilter on $\mathcal{B}_n(A)$…