Local mantles of $L[x]$

TL;DR

This paper investigates the structure of local mantles in models of set theory with large cardinals, showing they model ZFC + GCH + existence of a Woodin cardinal and are fully iterable strategy mice.

Contribution

It introduces a detailed analysis of the $oldsymbol{ ext{kappa}}$-mantle in models with Woodin cardinals, demonstrating it models ZFC + GCH + Woodin and is a fully iterable strategy mouse.

Findings

The $oldsymbol{ ext{kappa}}$-mantle models ZFC + GCH + Woodin cardinal.

The $oldsymbol{ ext{kappa}}$-mantle is a fully iterable strategy mouse.

Bounds on iteration strategies before adding $M_1^\#$.

Abstract

Assume ZFC. Let $\kappa$ be a cardinal. Recall that a ${<\kappa}$-ground is a transitive proper class $W$ modelling ZFC such that $V$ is a generic extension of $W$ via a forcing $\mathbb{P}\in W$ of cardinality ${<\kappa}$, and the $\kappa$-mantle is the intersection of all ${<\kappa}$-grounds. Assume there is a Woodin cardinal and a proper class of measurables, and let $x$ be a real of sufficiently high Turing degree. Let $\kappa$ be a limit cardinal of $L[x]$ of uncountable cofinality in $L[x]$. Using methods from Woodin's analysis of $\mathrm{HOD}^{L[x,G]}$, we analyze the $\kappa$-mantle of $L[x]$, and show that it models ZFC + GCH + "There is a Woodin cardinal". Moreover, we show that it is a fully iterable strategy mouse (analogous to $\mathrm{HOD}^{L[x,G]}$). We also analyze another form of "local mantle", partly assuming also a weak form of Turing determinacy. We also compute bounds on how much iteration strategy can be added to $M_1$ before $M_1^\#$ is added.