Open class determinacy is preserved by forcing

Abstract

The principle of open class determinacy is preserved by pre-tame class forcing, and after such forcing, every new class well-order is isomorphic to a ground-model class well-order. Similarly, the principle of elementary transfinite recursion ${\text{ETR}}_{\Gamma }$ for a fixed class well-order $\Gamma$ is preserved by pre-tame class forcing. The full principle ETR itself is preserved by countably strategically closed pre-tame class forcing, and after such forcing, every new class well-order is isomorphic to a ground-model class well-order. Meanwhile, it remains open whether ETR is preserved by all forcing, including the forcing merely to add a Cohen real.

14 pages. Questions and commentary can be posted at http://jdh.hamkins.org/open-class-determinacy-preserved-by-forcing

References in corpus (1)

• Satisfaction is not absolute