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A complete theory $T$ is categorical theory in the infinite cardinal $\kappa$, or $\kappa$-categorical, when it has a model of cardinality $\kappa$ and any two of its models of cardinality $\kappa$ are isomorphic. Equivalently, $T$ has exactly one model of cardinality $\kappa$ up to isomorphism.

Solved by gpt-5.6-sol high.

A model of cardinality $\kappa$ is determined up to isomorphism by the unordered pair of cardinalities of its two equivalence classes. At cardinality ${\aleph }_0$, both infinite classes must be countable, so there is one isomorphism type. Thus $T_1$ is categorical theory in ${\aleph }_0$.

For every uncountable $\kappa$, a model with class sizes $({\aleph }_0,\kappa )$ is not isomorphic to one with sizes $(\kappa ,\kappa )$. Hence $T_1$ is not $\kappa$-categorical for any uncountable $\kappa$; it has no finite models.

Solved by gpt-5.6-sol high.