Determinacy for open class games is preserved by forcing, CUNY Set Theory Seminar, April 2018

This will be a talk for the CUNY Set Theory Seminar, April 27, 2018, GC Room 6417, 10-11:45am (please note corrected date).

Abstract. 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.

5 thoughts on “Determinacy for open class games is preserved by forcing, CUNY Set Theory Seminar, April 2018

  1. When you say “forcing”, I assume you mean “set forcing”? You can easily break open determinacy with class forcing right? (i.e. open determinacy is a class forcing switch, not a button?).

    • No, our argument seems to apply to pre-tame class forcing, as well as set forcing. It seems to be quite generally preserved.

      Meanwhile, we don’t know if ETR is preserved, or other fragments of second-order set theory. For example, I’d like to show that global choice is conservative over GB+AC+ETR, and the most natural way to try to prove this would be to start with a model of this latter theory and then force global choice without adding sets. The problem, however, is that we don’t know that this forcing preserves ETR. The outline would work, however, with GB+AC+Open class determinacy.

  2. It seems that it’s quite difficult to keep track of what’s going on with introduced class-sized well-orders. This seemed to be the problem we encountered when looking at the variants of choice in second-order logic.

    • Yes, this was the original motivation for my question here. With open determinacy (and above), we now know that no new class well-orders are introduced, and perhaps this should be viewed as the main theorem here.

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