Open class determinacy is preserved by forcing

TL;DR

This paper demonstrates that open class determinacy and the principle of elementary transfinite recursion are preserved under certain types of pre-tame class forcing, with some cases remaining open for general forcing.

Contribution

It establishes preservation results for open class determinacy and ETR under pre-tame class forcing, extending understanding of their robustness in set-theoretic forcing.

Findings

Open class determinacy is preserved by pre-tame class forcing.

ETR for a fixed class well-order is preserved by pre-tame class forcing.

Full ETR is preserved by countably strategically closed pre-tame class 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.