Choiceless L\"owenheim-Skolem property and uniform definability of grounds

TL;DR

This paper explores the relationship between the L"owenheim-Skolem property, the definability of grounds, and the axiom of choice, revealing conditions under which grounds are uniformly definable and when the axiom of choice is forceable.

Contribution

It establishes a connection between the downward L"owenheim-Skolem property and the uniform definability of grounds without relying on the axiom of choice.

Findings

All grounds are uniformly definable under certain L"owenheim-Skolem conditions

The axiom of choice is forceable iff the universe is a small extension of a ZFC model

Provides conditions linking the axiom of choice and model extensions

Abstract

In this paper, without the axiom of choice, we show that if a certain downward L\"owenheim-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 $\mathsf{ZFC}$.