A premouse inheriting strong cardinals from V

Schlutzenberg Farmer

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, and iteration trees on L[E]^R lift to coarse iteration trees on R. We also prove that a slight weakening of (k+1)-condensation follows from (k,ω_1+1)-iterability in place of (k,ω_1,ω_1+1)-iterability. We also prove that full (k+1)-condensation follows from (k,ω_1+1)-iterability and (k+1)-solidity. We also prove general facts regarding generalizations of bicephali; these facts are needed in the proofs of the results above.