Varsovian models II

Sargsyan G, Schindler R, Schlutzenberg F

Abstract

Assume the existence of sufficent large cardinals. Let M_swn be theminimal iterable proper class L[E] model satisfying "there are δ_0 < κ_0 < δ_{n-1} < κ_{n-1} such that the δ_i are Woodin cardinals and the κ_i are strong cardinals". Let M = M_sw2 . We identify an inner model W_2^M of M, which is a proper class model satisfying "there are 2 Woodin cardinals", and is iterable both in the full set-theoretic universe V and in M, and closed under its own iteration strategy. The construction also yields significant information about the extent to which M knows its own iteration strategy. We characterizethe universe of W_2^M as the mantle and the least ground of M, and as HOD^{M[G]} for G ⊆ Coll(ω, λ) being M-generic with λ sufficiently large. These results correspond to facts already known for M_sw1, and the proofs are an elaboration on those, but there are substantial new issues and new methods used to handle them.