Karp complexity and classes with the independence property Original Research Article

A class View the MathML source of structures is controlled if for all cardinals λ, the relation of L∞,λ-equivalence partitions View the MathML source into a set of equivalence classes (as opposed to a proper class). We prove that no pseudo-elementary class with the independence property is controlled. By contrast, there is a pseudo-elementary class with the strict order property that is controlled (see Arch. Math. Logic 40 (2001) 69–88).