April 27
Joel David Hamkins,
CUNY
Determinacy for open class games is preserved by forcing
Open class determinacy is the principle of second order set theory asserting of every two-player game of perfect information, with plays coming from a (possibly proper) class $X$ and the winning condition determined by an open subclass of $X^\omega$, that one of the players has a winning strategy. This principle finds itself about midway up the hierarchy of second-order set theories between Gödel-Bernays set theory and Kelley-Morse, a bit stronger than the principle of elementary transfinite recursion ETR, which is equivalent to clopen determinacy, but weaker than GBC+$\Pi^1_1$-comprehension. In this talk, I’ll given an account of my recent joint work with W. Hugh Woodin, proving that open class determinacy is preserved by forcing. A central part of the proof is to show that in any forcing extension of a model of open class determinacy, every well-founded class relation in the extension is ranked by a ground model well-order relation. This work therefore fits into the emerging focus in set theory on the interaction of fundamental principles of second-order set theory with fundamental set theoretic tools, such as forcing. It remains open whether clopen determinacy or equivalently ETR is preserved by set forcing, even in the case of the forcing merely to add a Cohen real.