Karp complexity and classes with the independence property

by Laskowski and Shelah. [LwSh:687]
Annals Pure and Applied Logic, 2003
A class K of structures is controlled / if for all cardinals lambda, the relation of L_{infty, lambda}-equivalence partitions K 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.

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