J. D. Hamkins, On the strength of second-order set theories beyond ZFC, PSC-CUNY Research Award grant #69573-00 47, funded for 2016-2017.

**Abstract.** Professor Hamkins proposes to undertake research in the area of logic and foundations known as set theory, focused on the comparative strengths of several of the second-order set theories upon which several prominent recent research efforts have been based. These theories span the range from ZFC through GBC, plus various comprehension, transfinite recursion or class determinacy principles, up to KM and through the hierarchy to KM+ and beyond. Hamkins’s recent result with Gitman characterizing the precise strength of clopen determinacy for proper class games is a good start for the project, but many open questions remain, and Hamkins outlines various strategies that might solve them.

What is intended by “beyond KM+”? Any particular direction for strengthening this already powerful axiomatic system?

KM+ does not prove the class DC scheme, which asserts that if phi(X,Y) has the property that for every class X there is a class Y with phi(X,Y), then there is a class Z such that phi(Z_n,Z_{n+1}) for every n.