A model of cardinality is determined up to isomorphism by the unordered pair of cardinalities of its two equivalence classes. At cardinality , both infinite classes must be countable, so there is one isomorphism type. Thus is categorical theory in .
For every uncountable , a model with class sizes is not isomorphic to one with sizes . Hence is not -categorical for any uncountable ; it has no finite models.
The theory is not categorical in any infinite cardinal. In cardinality , one may add no infinite equivalence class or one countably infinite class; these models are not isomorphic. For uncountable , one can vary the number and cardinalities of infinite classes while retaining infinitely many classes of every finite size. These choices are isomorphism invariants.
Every finite partial isomorphism between models of extends by one point. If the new point belongs to a class already represented in the domain, choose an unused point in the corresponding target class. Otherwise choose an unused point in the other target class. Both classes are infinite, so the choice is always possible. The back-and-forth method shows that tuples with the same quantifier-free type have the same complete type. Therefore has quantifier elimination.
The formula saying that the -class of has exactly elements uses quantifiers and distinguishes elements in differently sized classes. No quantifier-free one-variable formula in the language can do so, since its only atomic information is and . Thus does not eliminate quantifiers.
Expand the language by unary predicates , one for each positive integer , interpreted as “the -class of has size ”. In this definitional expansion, back-and-forth on finite substructures gives quantifier elimination.
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