Jan Kruschewski | Searcharxiv

arXiv2025

Analysis of HOD for Admissible Structures

Let $n \geq 1$ and assume that there is a Woodin cardinal. For $x \in \mathbb{R}$ let $α_x$ be the least $β$ such that \[ L_β[x] \models Σ_n \text{-KP} + \exists κ(``κ\text{ is inaccessible and }κ^+ \text{ exists}"). \] We adapt the analysis of $\text{HOD}^{L[x,G]}$ as a strategy mouse to $L_{α_x}[x,G]$ for a cone of reals $x$. That is, we identify a mouse $\mathcal{M}^{\text{n-ad}}$ and define a class $H \subseteq L_{α_x}[x,G]$ as a natural analogue of $\text{HOD}^{L[x,G]} \subseteq L[x,G]$, and show that $H = M_\infty[Σ_0]$, where $M_\infty$ is an iterate of $\mathcal{M}^{\text{n-ad}}$ and $Σ_0$ a fragment of its iteration strategy.

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arXiv2022

On a Conjecture Regarding the Mouse Order for Weasels

We investigate Steel's conjecture in 'The Core Model Iterability Problem', that if $W$ and $R$ are $Ω+1$-iterable, $1$-small weasels, then $W\leq^{*}R$ iff there is a club $C\subsetΩ$ such that for all $α\in C$, if $α$ is regular, then the cardinal successor of $α$ in $W$ is less or equal than the cardinal successor of $α$ in $R$ . We will show that the conjecture fails, assuming that there is an iterable premouse which models $KP$ and which has a $Σ_{1}$-Woodin cardinal. On the other hand, we show that assuming there is no transitive model of $KP$ with a Woodin cardinal the conjecture holds. In the course of this we will also show that if $M$ is an iterable admissible premouse with a largest, regular, uncountable cardinal $δ$, and $\mathbb{P}$ is a forcing poset with the $δ$-c.c. in $M$, and $g$ is $M$-generic, but not necessarily $Σ_{1}$-generic, $M[g]$ is a model of $KP$. Moreover, if $M$ is such a mouse and $T$ is maximal normal iteration tree on $M$ such that $T$ is non-dropping on its main branch, then $M_{\infty}^{T}$ is again an iterable admissible premouse with a largest regular and uncountable cardinal. At last we answer another open question from 'The Core Model Iterability Problem' regarding the S-hull property.

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