[proofplan] We avoid comparing dimensions over a chosen parameter set. Instead, we use Morley's theorem: a countable complete theory categorical in one uncountable cardinal is $\omega$-stable. For countable $\omega$-stable theories, saturated models exist in every uncountable cardinal. Applying this existence theorem at the given cardinal $\kappa$ produces a saturated model of size $\kappa$, and categoricity identifies it with the unique model of size $\kappa$. [/proofplan] [step:Fix the categorical model of cardinality $\kappa$] Let $M \models T$ denote the unique model of cardinality $\kappa$. To prove that $M$ is $\kappa$-saturated, it suffices to show that some model of $T$ of cardinality $\kappa$ is $\kappa$-saturated: if $N \models T$ has cardinality $\kappa$ and is $\kappa$-saturated, then categoricity in $\kappa$ gives an isomorphism $f: N \to M$, and isomorphisms preserve realization of complete types over parameter sets of size less than $\kappa$. [/step] [step:Use uncountable categoricity to obtain $\omega$-stability] By Morley's uncountable categoricity theorem, a complete countable theory categorical in one uncountable cardinal is categorical in every uncountable cardinal and, in particular, is $\omega$-stable. The hypotheses required for this theorem are exactly present: $T$ is complete, the language of $T$ is countable, and $T$ is categorical in the uncountable cardinal $\kappa$. Hence $T$ is $\omega$-stable. [guided] The purpose of this step is to turn the categoricity hypothesis into a structural stability property strong enough to build saturated models. We invoke Morley's uncountable categoricity theorem. Its hypotheses are: the theory is complete, the language is countable, and the theory is categorical in some uncountable cardinal. These are precisely the assumptions on $T$ and $\kappa$ in the theorem statement. The conclusion we need is not the full transfer of categoricity to all uncountable cardinals, but the consequence that $T$ is $\omega$-stable. Thus, from uncountable categoricity of the complete countable theory $T$, we obtain that $T$ is $\omega$-stable. This replaces the earlier attempt to extract information from an omitted type over a chosen base; no parameter set is named, so there is no later invariance problem involving isomorphisms that fail to fix that base. [/guided] [/step] [step:Build a saturated model of size $\kappa$] We use the saturated-[model existence theorem](/theorems/1455) for countable $\omega$-stable theories: if $T$ is a complete countable $\omega$-stable theory and $\lambda$ is an uncountable cardinal, then there exists a $\lambda$-saturated model $N \models T$ of cardinality $\lambda$. The model-theoretic reason behind the theorem is that countable $\omega$-stability controls the number of complete types over parameter sets of size less than $\lambda$, so the usual elementary-chain construction can realize all such types while keeping the final model of cardinality $\lambda$. Applying this theorem with $\lambda := \kappa$ is legitimate because $\kappa$ is uncountable and the previous step shows that $T$ is $\omega$-stable. Therefore there exists a model $N \models T$ such that \begin{align*} |N| &= \kappa, \end{align*} and $N$ is $\kappa$-saturated. Here $\kappa$-saturation means the following: for every finite tuple of variables $x$, every subset $A \subset N$ with $|A| < \kappa$, and every complete type $p(x) \in S_x(A)$, where $S_x(A)$ denotes the set of complete types in the variables $x$ over the parameter set $A$, there is a tuple $b \in N^{|x|}$ such that $b \models p$. [/step] [step:Transfer saturation across the categorical isomorphism] Since $M \models T$, $N \models T$, and both models have cardinality $\kappa$, categoricity in $\kappa$ gives an isomorphism \begin{align*} f: N &\to M. \end{align*} Let $A \subset M$ be a parameter set with $|A| < \kappa$, let $x$ be a finite tuple of variables, and let $p(x) \in S_x(A)$ be a complete type in the variables $x$ over $A$. Define the inverse-image parameter set \begin{align*} A' := f^{-1}[A] \subset N. \end{align*} Then $|A'| = |A| < \kappa$. The isomorphism $f$ induces a bijection between $S_x(A')$ and $S_x(A)$ by replacing each parameter $c \in A'$ with $f(c) \in A$; let $p'(x) \in S_x(A')$ denote the preimage of $p(x)$ under this bijection. Because $N$ is $\kappa$-saturated, there is a tuple $b \in N^{|x|}$ such that $b \models p'$. Since elementary isomorphisms preserve truth of all formulas with parameters, the tuple $f(b) \in M^{|x|}$ realizes $p$. Thus every complete type in finitely many variables over every subset of $M$ of cardinality less than $\kappa$ is realized in $M$. [/step] [step:Conclude that the categorical model is saturated] The preceding step proves that the unique model $M \models T$ of cardinality $\kappa$ realizes every complete type in finitely many variables over every subset $A \subset M$ with $|A| < \kappa$. This is exactly $\kappa$-saturation of $M$. [/step]