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.
Categories: math.LO