Characterizing All Models in Infinite Cardinalities
Lauri Keskinen

Abstract:
Fix a cardinal kappa. We can ask the question what kind of a logic L is
needed to characterize all models of cardinality kappa (in a finite
vocabulary) up to isomorphism by their L-theories. In other words: for
which logics L it is true that if any models A and B satisfy the same
L-theory then they are isomorphic.
It is always possible to characterize models of cardinality kappa by their
L_{kappa ^+ ,kappa ^+ }-theories, but we are interested in finding a
``small" logic L, i.e. the sentences of L are hereditarily smaller than
kappa. For any cardinal kappa it is independent of ZFC whether any such
small definable logic L exists. If it exists it can be second order logic
for kappa=omega and fourth order logic or certain infinitary second order
logic L^2 _{kappa ,omega } for uncountable kappa. All models of cardinality
kappa can always be characterized by their theories in a small logic with
generalized quantifiers, but the logic may be not definable in the language
of set theory.