[proofplan] We count complete types over an arbitrary parameter set $A$ in each of the first three theories using their explicit descriptions: equality types are determined by equality to parameters or being new, algebraically closed field types are determined by algebraic data over the generated subfield or by transcendence, and vector-space types are determined by membership in the linear span or [linear independence](/page/Linear%20Independence) over it. These descriptions give at most $\kappa$ many complete types when $|A| \leq \kappa$ in the stated ranges. For dense linear orders, we exhibit a countable parameter set whose Dedekind cuts produce continuum many distinct complete $1$-types, contradicting $\aleph_0$-stability. [/proofplan] [step:Count equality types over a parameter set] Let $T_{=}$ denote the complete theory of an infinite set in the language whose only non-logical symbol is equality. Let $M \models T_{=}$, let $A \subset M$, and assume $|A| \leq \kappa$. A complete $1$-type over $A$ is determined by exactly one of the following alternatives: \begin{align*} &x = a \quad \text{for some } a \in A,\\ &x \neq a \quad \text{for every } a \in A. \end{align*} Thus \begin{align*} |S_1(A)| \leq |A| + 1 \leq \kappa. \end{align*} For an $n$-tuple $x=(x_1,\dots,x_n)$, a complete $n$-type over $A$ is determined by the equality pattern among the coordinates $x_i$ and the elements of $A$, together with the information of which coordinates, if any, are equal to parameters from $A$. For fixed finite $n$, the number of such patterns is bounded by \begin{align*} \sum_{m=0}^{n} C_{n,m}\, |A|^m \end{align*} for finite constants $C_{n,m} \in \mathbb N$ depending only on $n$. Since $\kappa$ is infinite and $|A| \leq \kappa$, this bound is at most $\kappa$. Hence $|S_n(A)| \leq \kappa$ for every $n \geq 1$, so $T_{=}$ is $\kappa$-stable. [guided] We first isolate what a type can say in pure equality. There are no relations, functions, or constants except the parameters from $A$, so a formula in one variable can only distinguish whether $x$ is equal to one of those parameters or different from all of them. Therefore a complete $1$-type over $A$ has exactly one of the following forms: \begin{align*} &\operatorname{tp}(a/A) \quad \text{for some } a \in A,\\ &\{x \neq a : a \in A\}. \end{align*} The first family has size $|A|$, and the second contributes one additional generic equality type. Hence \begin{align*} |S_1(A)| \leq |A|+1 \leq \kappa. \end{align*} For $n$-types, the same idea is applied to tuples. A complete type of $x=(x_1,\dots,x_n)$ records which coordinates are equal to which other coordinates, and which coordinates are equal to named parameters from $A$. Since $n$ is fixed, there are only finitely many equality partitions of the coordinate set $\{1,\dots,n\}$. For each block of such a partition, either that block is assigned to one parameter of $A$, or it is declared new and distinct from all parameters. Thus the number of choices is bounded by a finite sum of powers of $|A|$: \begin{align*} \sum_{m=0}^{n} C_{n,m}\, |A|^m, \end{align*} where $C_{n,m}$ is a finite number depending only on the possible equality patterns among $n$ coordinates. Because $\kappa$ is infinite, every finite power of a cardinal at most $\kappa$ is again at most $\kappa$. Hence $|S_n(A)| \leq \kappa$ for each finite $n$, which is precisely $\kappa$-stability for the equality theory. [/guided] [/step] [step:Count algebraically closed field types over the generated subfield] Let $T=\operatorname{ACF}_p$, where $p$ is either $0$ or a prime. Let $K \models T$, let $A \subset K$, and assume $|A| \leq \kappa$. Define $F_A \subset K$ to be the subfield generated by $A$ together with the prime field of $K$. Since $F_A$ is generated by at most $|A|+\aleph_0$ elements, \begin{align*} |F_A| \leq |A|+\aleph_0 \leq \kappa. \end{align*} A complete $1$-type over $A$ is determined either by the [minimal polynomial](/page/Minimal%20Polynomial) of an algebraic element over $F_A$, or by the transcendental type over $F_A$. The set $F_A[X]$ of one-variable polynomials over $F_A$ has cardinality \begin{align*} |F_A[X]| = |F_A|+\aleph_0 \leq \kappa, \end{align*} so there are at most $\kappa$ algebraic $1$-types and one transcendental $1$-type. Hence $|S_1(A)| \leq \kappa$. For an $n$-tuple, the same description is obtained by choosing a transcendence basis subtuple over $F_A$ and then recording algebraic data for the remaining coordinates over the finitely generated [field extension](/page/Field%20Extension) generated by that subtuple. For fixed $n$, the number of choices of the algebraic data is bounded by the number of finite polynomial systems over $F_A$, which is \begin{align*} |F_A|+\aleph_0 \leq \kappa. \end{align*} Therefore $|S_n(A)| \leq \kappa$ for every finite $n$, and $\operatorname{ACF}_p$ is $\kappa$-stable for every infinite $\kappa$. [/step] [step:Count vector-space types over the span of the parameters] Let $T_F$ denote the complete theory of infinite-dimensional vector spaces over a fixed field $F$, in the usual language of $F$-vector spaces. Let $V \models T_F$, let $A \subset V$, and assume $|A| \leq \kappa$ with $\kappa \geq |F|+\aleph_0$. Define $W_A \subset V$ to be the $F$-linear span of $A$. Each element of $W_A$ is a finite $F$-linear combination of elements of $A$, so \begin{align*} |W_A| \leq \sum_{m=0}^{\infty} |F|^m |A|^m \leq \max\{|F|,|A|,\aleph_0\} \leq \kappa. \end{align*} A complete $1$-type over $A$ is determined either by the assertion $x=w$ for some $w \in W_A$, or by the assertion that $x$ is linearly independent over $W_A$. Hence \begin{align*} |S_1(A)| \leq |W_A|+1 \leq \kappa. \end{align*} For an $n$-tuple $x=(x_1,\dots,x_n)$, its type over $A$ is determined by the finite-dimensional linear dependence pattern of $x_1,\dots,x_n$ over $W_A$, together with the specific values in $W_A$ of all dependent linear combinations. For fixed $n$, this data is bounded by finitely many choices of coefficients from $F$ and finitely many choices of vectors from $W_A$, so the total number of possibilities is at most \begin{align*} \max\{|F|,|W_A|,\aleph_0\} \leq \kappa. \end{align*} Thus $|S_n(A)| \leq \kappa$ for every finite $n$, and $T_F$ is $\kappa$-stable whenever $\kappa \geq |F|+\aleph_0$. [guided] The parameter set $A$ does not matter directly in a [vector space](/page/Vector%20Space); what matters is the subspace it spans. Define $W_A$ to be the $F$-linear span of $A$ inside $V$. Every vector in $W_A$ has the form \begin{align*} \lambda_1 a_1+\cdots+\lambda_m a_m, \end{align*} where $m \in \mathbb N$, $\lambda_1,\dots,\lambda_m \in F$, and $a_1,\dots,a_m \in A$. Therefore the number of possible vectors in $W_A$ is bounded by the number of finite choices of coefficients and parameters: \begin{align*} |W_A| \leq \sum_{m=0}^{\infty} |F|^m |A|^m. \end{align*} Since $\kappa$ is infinite, $|A| \leq \kappa$, and $\kappa \geq |F|+\aleph_0$, this cardinal is at most $\kappa$. Now consider a single variable $x$. In the theory of vector spaces, the formulas over $A$ can only test finite linear equations with coefficients in $F$ and constants from $W_A$. Thus there are two possibilities. Either $x$ is equal to a specific vector $w \in W_A$, or $x$ is not in $W_A$, in which case it realizes the unique type saying that $x$ is linearly independent over $W_A$. Hence \begin{align*} |S_1(A)| \leq |W_A|+1 \leq \kappa. \end{align*} For a tuple $x=(x_1,\dots,x_n)$, the same principle applies with finite-dimensional linear algebra. The type records which finite $F$-linear combinations of the coordinates land in $W_A$, and what their values in $W_A$ are. There are only finitely many coordinates, so this information is controlled by finitely many coefficients from $F$ and finitely many target values from $W_A$. Therefore the number of possible complete $n$-types is bounded by \begin{align*} \max\{|F|,|W_A|,\aleph_0\} \leq \kappa. \end{align*} This gives $|S_n(A)| \leq \kappa$ for every finite $n$, which is the desired stability bound. [/guided] [/step] [step:Produce continuum many order types over a countable dense set] Let $T_{\operatorname{DLO}}$ denote the complete theory of dense linear orders without endpoints. Let $M \models T_{\operatorname{DLO}}$, and choose a countable subset $A \subset M$ order-isomorphic to $\mathbb Q$. For each Dedekind cut $C \subset A$, define a partial $1$-type $p_C(x)$ over $A$ by \begin{align*} p_C(x) = \{a < x : a \in C\} \cup \{x < b : b \in A \setminus C\}, \end{align*} where $C$ is downward closed and has no greatest element in $A$, while $A \setminus C$ is upward closed and has no least element in $A$. Every finite subset of $p_C(x)$ is satisfiable in $M$ because $M$ is a dense linear order without endpoints and $A$ has the order type of $\mathbb Q$. By compactness, $p_C(x)$ extends to a complete $1$-type over $A$. If $C \neq D$ are distinct cuts, choose $a \in A$ lying in the symmetric difference $C \triangle D$. Then the formula $a < x$ belongs to exactly one of $p_C$ and $p_D$, so the corresponding complete types are distinct. The set $\mathbb Q$ has $2^{\aleph_0}$ Dedekind cuts; therefore \begin{align*} |S_1(A)| \geq 2^{\aleph_0} > \aleph_0. \end{align*} Since $A$ is countable, $T_{\operatorname{DLO}}$ is not $\aleph_0$-stable. [/step] [step:Combine the four counts into the dichotomy] The preceding steps prove that, over every parameter set of cardinality at most $\kappa$, the theories of pure equality and algebraically closed fields have at most $\kappa$ complete finite types for every infinite $\kappa$. They also prove that infinite-dimensional $F$-vector spaces have at most $\kappa$ complete finite types whenever $\kappa \geq |F|+\aleph_0$. The dense linear order example gives a countable parameter set over which there are $2^{\aleph_0}$ complete $1$-types, so it violates the defining bound for $\aleph_0$-stability. This proves all four asserted stability behaviours. [/step]