Geology of symmetric grounds

TL;DR

This paper explores the structure of symmetric grounds in set-theoretic geology, demonstrating their definability and downward directedness under certain assumptions and forcing conditions.

Contribution

It introduces the concept of symmetric grounds, proves their uniform definability, and establishes their downward directedness when the Axiom of Choice is forceable.

Findings

Symmetric grounds are uniformly definable under certain assumptions.

Symmetric grounds are downward directed if the Axiom of Choice is forceable.

The paper advances understanding of the structure of symmetric grounds in set theory.

Abstract

Let us say that a model of $\mathsf{ZF}$ is a symmetric ground if $V$ is a symmetric extension of the model. In this paper, we investigate set-theoretic geology of symmetric grounds. Under a certain assumption, we show that all symmetric grounds of $V$ are uniformly definable. We also show that if $\mathsf{AC}$ is forceable over $V$, then the symmetric grounds are downward directed.