On Separating Wholeness Axioms
Hanul Jeon
TL;DR
The paper establishes a strict separation in the Wholeness hierarchy by proving that for each $n\ge 0$, $\mathsf{ZFC}+\mathsf{WA}_{n+1}$ proves the consistency of $\mathsf{ZFC}+\mathsf{WA}_n$, while also showing that $\mathsf{ZFC}+\mathsf{WA}_n$ is finitely axiomatizable but $\mathsf{ZFC}+\mathsf{WA}$ is not. It develops a formal framework combining a $j$-augmented Levy-Fleischmann hierarchy with partial truth predicates and a cut-free sequent calculus to carry out consistency proofs. The results demonstrate a clear, stepwise increase in consistency strength along the WA_n ladder and extend the method to separations involving $\Pi^j_n$-Induction, thereby clarifying the structure and axiomatizability of the Wholeness hierarchy. The approach provides a proof-theoretic path to separating large-cardinal-like axioms within ZFC and illustrates the utility of partial truth predicates in establishing metatheoretical consistency results.
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.