[proofplan] We use Morley's Ehrenfeucht-Mostowski order-spectrum theorem for a countable complete theory with the order property. The theorem assigns to each linear order $I$ a model $M_I \models T$ of size $|I|+\aleph_0$ containing an order-indiscernible skeleton indexed by $I$, and the formula witnessing the order property recovers the order relation on that skeleton. We define the oriented cut invariant used by the theorem, choose two linear orders of cardinality $\kappa$ with different values of that invariant, and then use the theorem's isomorphism-invariance clause to separate the resulting models. [/proofplan]

[step:Apply the order-spectrum form of the Ehrenfeucht-Mostowski construction]Fix an uncountable cardinal $\kappa$. Let $\varphi(x;y)$ be a formula witnessing the [order property](/page/Order%20Property) for $T$. We first specify the invariant used below. If $L$ is a linear order, a cut of $L$ is a pair $(A,B)$ of subsets of $L$ such that $A \cup B=L$, every element of $A$ is $<_L$ every element of $B$, and $A$ is an initial segment of $L$. We allow endpoint cuts, so $A=\varnothing$ or $B=\varnothing$ is permitted. For a subset $A \subset L$, let $\operatorname{cf}_L(A)$ denote the least cardinality of a subset $C \subset A$ that is cofinal in $A$ with respect to $<_L$, with value $0$ when $A=\varnothing$. For a subset $B \subset L$, let $\operatorname{ci}_L(B)$ denote the least cardinality of a subset $D \subset B$ that is coinitial in $B$ with respect to $<_L$, with value $0$ when $B=\varnothing$. Define the oriented cut spectrum \begin{align*} \operatorname{Cuts}(L) := \{(\operatorname{cf}_L(A),\operatorname{ci}_L(B)) : (A,B) \text{ is a cut of } L\}. \end{align*} Here $\operatorname{cf}(\kappa)$ denotes the cofinality of the cardinal $\kappa$, namely the least cardinality of a cofinal subset of the ordinal $\kappa$. We use the following standard result, [Morley's order-spectrum theorem](/page/Morley%20Order-Spectrum%20Theorem), in its Ehrenfeucht-Mostowski form. If $T$ is a complete countable theory and $\varphi(x;y)$ has the order property, then for every linear order $I$ there is a model $M_I \models T$ and a sequence \begin{align*} s_I: I &\to M_I^{|x|} \times M_I^{|y|} \\ i &\mapsto (a_i,b_i) \end{align*} such that each $a_i$ is a tuple from $M_I^{|x|}$ matching the arity of $x$, and each $b_i$ is a tuple from $M_I^{|y|}$ matching the arity of $y$. The theorem gives: 1. the sequence $(a_i,b_i)_{i \in I}$ is [order-indiscernible](/page/Indiscernible%20Sequence) in $M_I$; 2. for all $i,j \in I$, \begin{align*} M_I \models \varphi(a_i;b_j) \quad \Longleftrightarrow \quad i <_I j; \end{align*} 3. $|M_I| = |I|+\aleph_0$; 4. if $I$ and $J$ are linear orders and $M_I \cong M_J$ as $T$-models, then $\operatorname{Cuts}(I)=\operatorname{Cuts}(J)$. This is Morley's Ehrenfeucht-Mostowski order-spectrum theorem for formulas with the order property. Clause 4 is the precise invariant-preservation statement used below: it is not obtained merely from preservation of $\varphi$ on one named skeleton, but from the EM order-spectrum analysis showing that the oriented cut spectrum coded by the $\varphi$-order is invariant under arbitrary isomorphisms of the resulting $T$-reducts. Since $\kappa$ is uncountable, any linear order $I$ with $|I|=\kappa$ gives \begin{align*} |M_I| = |I|+\aleph_0 = \kappa. \end{align*} Thus it remains to choose two linear orders of cardinality $\kappa$ with different cut spectra.[/step]

[guided]The order property says that one formula can encode arbitrarily long finite linear orders. [Morley's order-spectrum theorem](/page/Morley%20Order-Spectrum%20Theorem) upgrades this finite coding into models built over arbitrary linear orders, and its preservation clause says that the oriented cut spectrum of the indexing order is recoverable from the resulting $T$-model up to isomorphism. Fix an uncountable cardinal $\kappa$ and a formula $\varphi(x;y)$ witnessing the order property. For a linear order $L$, a cut is a decomposition $(A,B)$ where $A \cup B=L$, every element of $A$ is less than every element of $B$, and $A$ is an initial segment. We include endpoint cuts. The invariant is \begin{align*} \operatorname{Cuts}(L) := \{(\operatorname{cf}_L(A),\operatorname{ci}_L(B)) : (A,B) \text{ is a cut of } L\}, \end{align*} where $\operatorname{cf}_L(A)$ is the least size of a cofinal subset of $A$, $\operatorname{ci}_L(B)$ is the least size of a coinitial subset of $B$, and the value is $0$ for the empty side. The hypotheses of Morley's order-spectrum theorem are satisfied: $T$ is complete and countable by assumption, and $\varphi(x;y)$ has the order property by choice of $\varphi$. Therefore, for each linear order $I$, the theorem gives a model $M_I \models T$ and a map \begin{align*} s_I: I &\to M_I^{|x|} \times M_I^{|y|} \\ i &\mapsto (a_i,b_i), \end{align*} where $a_i$ is a tuple from $M_I^{|x|}$ and $b_i$ is a tuple from $M_I^{|y|}$. The sequence is order-indiscernible, and $\varphi$ reads the indexing order: \begin{align*} M_I \models \varphi(a_i;b_j) \quad \Longleftrightarrow \quad i <_I j. \end{align*} The theorem also gives $|M_I|=|I|+\aleph_0$ and the preservation clause: if $M_I \cong M_J$ as $T$-models, then \begin{align*} \operatorname{Cuts}(I)=\operatorname{Cuts}(J). \end{align*} This is the key point: an arbitrary isomorphism need not carry the chosen skeleton of one EM model to the chosen skeleton of the other, so the cut-spectrum preservation is a genuine theorem, not a consequence of elementwise preservation of $\varphi$ alone. Now choose $I=\kappa$ with its usual well-order and choose $J=\kappa^*$ with the reverse order. Both have cardinality $\kappa$, so the size clause gives \begin{align*} |M_I|=|M_J|=\kappa+\aleph_0=\kappa. \end{align*} For $I=\kappa$, every cut is determined by an ordinal $\alpha \leq \kappa$: \begin{align*} A_\alpha &= \{\beta < \kappa : \beta < \alpha\}, & B_\alpha &= \{\beta < \kappa : \alpha \leq \beta\}. \end{align*} If $\alpha<\kappa$, then $B_\alpha$ has least element $\alpha$, so $\operatorname{ci}_I(B_\alpha)=1$; if $\alpha=\kappa$, then $B_\kappa=\varnothing$ and $\operatorname{cf}_I(A_\kappa)=\operatorname{cf}(\kappa)$. Hence \begin{align*} (\operatorname{cf}(\kappa),0) \in \operatorname{Cuts}(I). \end{align*} For $J=\kappa^*$, every nonempty initial segment has a greatest element in the order $J$: it is of the form $\{\beta<\kappa: \alpha \leq \beta\}$ for some $\alpha<\kappa$, and $\alpha$ is greatest with respect to $<_J$. Therefore every cut of $J$ with nonempty lower side has first coordinate $1$, while the left endpoint cut has first coordinate $0$. Thus the first coordinate of every pair in $\operatorname{Cuts}(J)$ is either $0$ or $1$. Since $\kappa$ is uncountable, $\operatorname{cf}(\kappa)>1$, so \begin{align*} (\operatorname{cf}(\kappa),0) \notin \operatorname{Cuts}(J). \end{align*} Therefore $\operatorname{Cuts}(I) \neq \operatorname{Cuts}(J)$. If there were an isomorphism $F:M_I\to M_J$, Morley's preservation clause would imply $\operatorname{Cuts}(I)=\operatorname{Cuts}(J)$, contradicting the computation above. Hence $M_I$ and $M_J$ are non-isomorphic models of $T$ of cardinality $\kappa$. Since $\kappa$ was arbitrary, every uncountable cardinal supports at least two non-isomorphic models of $T$.[/guided]

[step:Choose two linear orders of size $\kappa$ with different cut spectra] Let $I$ be the ordinal order $\kappa$ with its usual well-ordering. Let $J$ be the reverse ordinal order $\kappa^*$. Both $I$ and $J$ have cardinality $\kappa$. We compare the oriented cut spectra using the endpoint convention fixed above. In $I=\kappa$, the left endpoint cut $(\varnothing,I)$ contributes \begin{align*} (0,1) \in \operatorname{Cuts}(I), \end{align*} because $I$ has a least element, so the singleton containing that least element is coinitial in $I$. The right endpoint cut $(I,\varnothing)$ contributes \begin{align*} (\operatorname{cf}(\kappa),0) \in \operatorname{Cuts}(I), \end{align*} because a subset of the ordinal $\kappa$ is cofinal in $I$ exactly when it is cofinal in the cardinal $\kappa$. In $J=\kappa^*$, we now rule out the pair $(\operatorname{cf}(\kappa),0)$ from all cuts, not only endpoint cuts. Let $(A,B)$ be any cut of $J$. If $A=\varnothing$, then $\operatorname{cf}_J(A)=0$. If $A\neq\varnothing$, then $A$ is an initial segment of the reverse well-order $\kappa^*$, so there is an ordinal $\alpha<\kappa$ such that \begin{align*} A=\{\beta<\kappa : \alpha \leq \beta\}. \end{align*} With respect to the reverse order $<_J$, the element $\alpha$ is greatest in $A$. Hence the singleton $\{\alpha\}$ is cofinal in $A$, and \begin{align*} \operatorname{cf}_J(A)=1. \end{align*} Thus the first coordinate of every pair in $\operatorname{Cuts}(J)$ is either $0$ or $1$. Since $\kappa$ is uncountable, $\operatorname{cf}(\kappa)>1$. Therefore \begin{align*} (\operatorname{cf}(\kappa),0) \notin \operatorname{Cuts}(J), \end{align*} while the right endpoint cut of $I=\kappa$ showed \begin{align*} (\operatorname{cf}(\kappa),0) \in \operatorname{Cuts}(I). \end{align*} Consequently $\operatorname{Cuts}(I) \neq \operatorname{Cuts}(J)$. By the order-spectrum theorem stated above, this difference is reflected in the first-order traces of the $\varphi$-order inside $M_I$ and $M_J$. [/step]

[step:Separate the Ehrenfeucht-Mostowski models by the preserved cut invariant] The models $M_I$ and $M_J$ both satisfy $T$ and both have cardinality $\kappa$. Suppose, toward a contradiction, that there is an isomorphism \begin{align*} F: M_I &\to M_J. \end{align*} By the isomorphism-invariance clause in Morley's order-spectrum theorem, an isomorphism $F: M_I \to M_J$ would imply \begin{align*} \operatorname{Cuts}(I)=\operatorname{Cuts}(J). \end{align*} This is the exact point at which the full order-spectrum theorem is used: preservation of truth for $\varphi(x;y)$ alone would only say that $F$ preserves the $\varphi$-relation on images of skeleton elements, whereas the theorem asserts that the oriented cut spectrum coded by the EM construction is invariant under arbitrary isomorphisms of the resulting $T$-models. This contradicts the established inequality $\operatorname{Cuts}(I) \neq \operatorname{Cuts}(J)$ for $I=\kappa$ and $J=\kappa^*$. Therefore no such isomorphism $F$ exists. [/step]

[step:Conclude that every uncountable cardinal supports at least two models] We have constructed two models $M_I,N:=M_J \models T$ such that \begin{align*} |M_I| = |M_J| = \kappa \end{align*} and $M_I \not\cong M_J$. Since $\kappa$ was an arbitrary uncountable cardinal, for every uncountable cardinal $\kappa$ the theory $T$ has at least two non-isomorphic models of cardinality $\kappa$. [/step]