 Solution

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.