Open determinacy for class games

TL;DR

This paper explores the limits of open determinacy for class games within set theory, showing its unprovability in ZFC and GBC, but provability in a stronger theory, and linking it to truth and recursion principles.

Contribution

It establishes the unprovability of open determinacy for class games in ZFC and GBC, and connects it to principles of truth, recursion, and stronger set theories.

Findings

Open determinacy implies the existence of satisfaction classes for truth.

Clopen determinacy is equivalent to elementary transfinite recursion over well-founded class relations.

Open determinacy is provable in GBC plus a1^1_1d-comprehension.

Abstract

The principle of open determinacy for class games---two-player games of perfect information with plays of length $\omega$, where the moves are chosen from a possibly proper class, such as games on the ordinals---is not provable in Zermelo-Fraenkel set theory ZFC or G\"odel-Bernays set theory GBC, if these theories are consistent, because provably in ZFC there is a definable open proper class game with no definable winning strategy. In fact, the principle of open determinacy and even merely clopen determinacy for class games implies Con(ZFC) and iterated instances Con(Con(ZFC)) and more, because it implies that there is a satisfaction class for first-order truth and indeed a transfinite tower of truth predicates for iterated truth-about-truth, relative to any class parameter. This is perhaps explained, in light of the Tarskian recursive definition of truth, by the more general fact that the principle of clopen determinacy is exactly equivalent over GBC to the principle of elementary transfinite recursion ETR over well-founded class relations. Meanwhile, the principle of open determinacy for class games is provable in the stronger theory GBC+$\Pi^1_1$-comprehension, a proper fragment of Kelley-Morse set theory KM.