Choiceless Löwenheim-Skolem property and uniform definability of grounds

Abstract

In this paper, without the axiom of choice, we show that if a certain downward Löwenheim-Skolem property holds then all grounds are uniformly definable. We also prove that the axiom of choice is forceable if and only if the universe is a small extension of some transitive model of $\mathsf{ZFC}$.