A premouse inheriting strong cardinals from $V$

TL;DR

This paper constructs a premouse inner model that inherits all strong and Woodin cardinals from any universe modeling ZFC, and explores the implications of various condensation and solidity principles in inner model theory.

Contribution

It introduces a new premouse inner model $L[ ext{E}]$ inheriting large cardinals and develops the theory of bicephali to analyze condensation and solidity.

Findings

The inner model $L[ ext{E}]$ inherits all strong and Woodin cardinals.

$(k+1)$-condensation follows from $(k+1)$-solidity and $(k, ext{ω}_1+1)$-iterability.

A weakened form of $(k+1)$-condensation is derived from $(k, ext{ω}_1+1)$-iterability.

Abstract

We identify a premouse inner model $L[\mathbb{E}]$, such that for any coarsely iterable background universe $R$ modelling $\mathrm{ZFC}$, $L[\mathbb{E}]^R$ is a proper class premouse of $R$ inheriting all strong and Woodin cardinals from $R$. Moreover, for each $\alpha\in\mathrm{OR}$, $L[\mathbb{E}]^R|\alpha$ is $(\omega,\alpha)$-iterable, via iteration trees which lift to coarse iteration trees on $R$. We prove that $(k+1)$-condensation follows from $(k+1)$-solidity together with $(k,\omega_1+1)$-iterability (that is, roughly, iterability with respect to normal trees). We also prove that a slight weakening of $(k+1)$-condensation follows from $(k,\omega_1+1)$-iterability (without the $(k+1)$-solidity hypothesis). The results depend on the theory of generalizations of bicephali, which we also develop.