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The countable dense linear order is unique up to a non-unique isomorphism. The countable atomless Boolean algebra is unique up to a non-unique isomorphism. The Rando graph is unique up to a non-unique isomorphism. (Algebraically closed fields of a given characteristic and transcendence degree, and vector spaces of given dimension over a given field, are other examples already mentioned above.)

In general, if $T$ is a $\kappa$-categorical first-order theory (in a countable language), then the model $M$ of $T$ of cardinality $\kappa$ is unique up to a non-unique isomorphism.

Even more generally: if $T$ is any complete theory and $\kappa$ an infinite cardinal, then the saturated model $M$ of $T$ of cardinality $\kappa$—if it exists at all—is unique up to a non-unique isomorphism. ($M$ is unique by a standard back-and-forth argument. Non-uniqueness of the isomorphism amounts to saying that $\operatorname{Aut}(M)$ is nontrivial. By homogeneity, it suffices to exhibit two elements of $M$ with the same type. If there exists a nonprincipal parameter-free $1$-type, we can easily find two elements that realize it. If all $1$-types are principal, there are only finitely many, hence two elements of $M$ have to realize the same type by the pigeonhole principle.)

The countable dense linear order is unique up to a non-unique isomorphism. The countable atomless Boolean algebra is unique up to a non-unique isomorphism. The random graph is unique up to a non-unique isomorphism. (Algebraically closed fields of a given characteristic and transcendence degree, and vector spaces of given dimension over a given field, are other examples already mentioned above.)

In general, if $T$ is a $\kappa$-categorical first-order theory (in a countable language), then the model $M$ of $T$ of cardinality $\kappa$ is unique up to a non-unique isomorphism.

Even more generally: if $T$ is any complete theory and $\kappa$ an infinite cardinal, then the saturated model $M$ of $T$ of cardinality $\kappa$—if it exists at all—is unique up to a non-unique isomorphism. ($M$ is unique by a standard back-and-forth argument. Non-uniqueness of the isomorphism amounts to saying that $\operatorname{Aut}(M)$ is nontrivial. By homogeneity, it suffices to exhibit two elements of $M$ with the same type. If there exists a nonprincipal parameter-free $1$-type, we can easily find two elements that realize it. If all $1$-types are principal, there are only finitely many, hence two elements of $M$ have to realize the same type by the pigeonhole principle.)

The countable dense linear order is unique up to a non-unique isomorphism. The countable atomless Boolean algebra is unique up to a non-unique isomorphism. The random graph is unique up to a non-unique isomorphism. (Algebraically closed fields of a given characteristic and transcendence degree, and vector spaces of given dimension over a given field, are other examples already mentioned above.)

In general, if $T$ is a $\kappa$-categorical first-order theory (in a countable language), then the model $M$ of $T$ of cardinality $\kappa$ is unique up to a non-unique isomorphism.

Even more generally: if $T$ is any complete theory and $\kappa$ an infinite cardinal, then the saturated model $M$ of $T$ of cardinality $\kappa$—if it exists at all—is unique up to a non-unique isomorphism. ($M$ is unique by a standard back-and-forth argument. Non-uniqueness of the isomorphism amounts to saying that $\operatorname{Aut}(M)$ is nontrivial. By homogeneity, it suffices to exhibit two elements of $M$ with the same type. If there exists a nonprincipal parameter-free $1$-type, we can easily find two elements that realize it. If all $1$-types are principal, there are only finitely many, hence two elements of $M$ have to realize the same type by the pigeonhole principle.)

The countable dense linear order is unique up to a non-unique isomorphism. The countable atomless Boolean algebra is unique up to a non-unique isomorphism. The Rando graph is unique up to a non-unique isomorphism. (Algebraically closed fields of a given characteristic and transcendence degree, and vector spaces of given dimension over a given field, are other examples already mentioned above.)

In general, if $T$ is a $\kappa$-categorical first-order theory (in a countable language), then the model $M$ of $T$ of cardinality $\kappa$ is unique up to a non-unique isomorphism.

Even more generally: if $T$ is any complete theory and $\kappa$ an infinite cardinal, then the saturated model $M$ of $T$ of cardinality $\kappa$—if it exists at all—is unique up to a non-unique isomorphism. ($M$ is unique by a standard back-and-forth argument. Non-uniqueness of the isomorphism amounts to saying that $\operatorname{Aut}(M)$ is nontrivial. By homogeneity, it suffices to exhibit two elements of $M$ with the same type. If there exists a nonprincipal parameter-free $1$-type, we can easily find two elements that realize it. If all $1$-types are principal, there are only finitely many, hence two elements of $M$ have to realize the same type by the pigeonhole principle.)

The countable dense linear order is unique up to a non-unique isomorphism. The countable atomless Boolean algebra is unique up to a non-unique isomorphism. (Algebraically closed fields of a given characteristic and transcendence degree, and vector spaces of given dimension over a given field, are other examples already mentioned above.)

In general, if $T$ is any $\kappa$-categorical first-order theory (in a countable language), then the model $M$ of $T$ of cardinality $\kappa$ is unique up to a non-unique isomorphism. (This amounts to stating that $\operatorname{Aut}(M)$ is nontrivial. This holds e.g. because $M$ is saturated, and the number of its parameter-free $1$-types is strictly less than $\kappa$, hence there are two elements of $M$ with the same type, and one can find an automorphism mapping one to the other by homogeneity.)

The countable dense linear order is unique up to a non-unique isomorphism. The countable atomless Boolean algebra is unique up to a non-unique isomorphism. The random graph is unique up to a non-unique isomorphism. (Algebraically closed fields of a given characteristic and transcendence degree, and vector spaces of given dimension over a given field, are other examples already mentioned above.)

In general, if $T$ is a $\kappa$-categorical first-order theory (in a countable language), then the model $M$ of $T$ of cardinality $\kappa$ is unique up to a non-unique isomorphism.

Even more generally: if $T$ is any complete theory and $\kappa$ an infinite cardinal, then the saturated model $M$ of $T$ of cardinality $\kappa$—if it exists at all—is unique up to a non-unique isomorphism. ($M$ is unique by a standard back-and-forth argument. Non-uniqueness of the isomorphism amounts to saying that $\operatorname{Aut}(M)$ is nontrivial. By homogeneity, it suffices to exhibit two elements of $M$ with the same type. If there exists a nonprincipal parameter-free $1$-type, we can easily find two elements that realize it. If all $1$-types are principal, there are only finitely many, hence two elements of $M$ have to realize the same type by the pigeonhole principle.)