A class K of structures is controlled if for all cardinals λ, the relation of L∞,λ-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 (see Arch. Math. Logic 40 (2001) 69-88). © 2002 Elsevier Science B.V. All rights reserved.
Laskowski, M. C., & Shelah, S. (2003). Karp complexity and classes with the independence property. Annals of Pure and Applied Logic, 120(1–3), 263–283. https://doi.org/10.1016/S0168-0072(02)00080-5