The $\omega$-th inner mantle

TL;DR

This paper explores complex behaviors of the inner mantle hierarchy at the first limit stage, revealing that the $oldsymbol{ extomega}$-th inner mantle can be non-definable or fail to satisfy the Axiom of Choice, answering open questions.

Contribution

It demonstrates that the $oldsymbol{ extomega}$-th inner mantle can be non-definable or violate AC, providing new insights into the structure of inner models.

Findings

The $oldsymbol{ extomega}$-th inner mantle can be non-definable.

The $oldsymbol{ extomega}$-th inner mantle can be definable but fail AC.

Answers open questions by Fuchs, Hamkins, and Reitz.

Abstract

This article investigates pathological behavior at the first limit stage in the sequence of inner mantles, obtained by iterating the definition of the mantle to get smaller and smaller inner models. I show: (A) it is possible that the $\omega$-th inner mantle is not a definable class; and (B) it is possible that the $\omega$-th inner mantle is a definable class but does not satisfy $\mathsf{AC}$. This answers a pair of questions of Fuchs, Hamkins, and Reitz [FHR15].