This probably has to do with the (upward) Löwenheim–Skolem Theorem, according to which Γ has models of every infinite cardinality. Then the question reduces to showing that the class containing all cardinals is bigger than any set.
Explain me please, why if the set of first-order sentences has an infinite model, then the set (class) of all models of , , is strictly bigger than any set ?
Here is a passage from a textbook:
... is sometimes unbounded. It is unbounded precisely when has an infinite model. By unbounded we mean that for any set , is strictly bigger than ...
This probably has to do with the (upward) Löwenheim–Skolem Theorem, according to which Γ has models of every infinite cardinality. Then the question reduces to showing that the class containing all cardinals is bigger than any set.