On Separating Wholeness Axioms

TL;DR

This paper investigates the relationships between different wholeness axioms in set theory, proving implications of consistency and axiomatizability properties among them.

Contribution

It establishes that higher wholeness axioms imply the consistency of lower ones and shows that the base theory with wholeness axioms is finitely axiomatizable, unlike the full theory.

Findings

$ ext{ZFC+WA}_{n+1}$ implies the consistency of $ ext{ZFC+WA}_n$ for all $n",

$ ext{ZFC+WA}_n$ is finitely axiomatizable

$ ext{ZFC+WA}$ is not finitely axiomatizable

Abstract

In this paper, we prove that $\mathsf{ZFC+WA}_{n+1}$ implies the consistency of $\mathsf{ZFC+WA}_n$ for $n\ge 0$. We also prove that $\mathsf{ZFC+WA}_n$ is finitely axiomatizable, and $\mathsf{ZFC+WA}$ is not finitely axiomatizable.