A premouse inheriting strong cardinals from V

Abstract

We identify a premouse inner model L[E], such that for any coarsely iterable background universe R modelling ZFC, L[E]R is a proper class premouse of R inheriting all strong and Woodin cardinals from R. For each ordinal α, L[E]R|α is (ω,α)-iterable, via iteration trees which lift to coarse iteration trees on R.We prove that (k+1)-condensation follows from (k+1)-solidity and (k,ω1+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,ω1+1)-iterability (without the (k+1)-solidity hypothesis).The results depend on the theory of generalizations of bicephali, which we also develop.