# Geology of symmetric grounds

**Authors:** Toshimichi Usuba  
**arXiv:** [1912.10246](https://arxiv.org/abs/1912.10246)  
**Date:** 2019-12-21  
**Summary page:** https://www.summarizepaper.com/en/paper/1912.10246v3/

## 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.