SUITABLE EXTENDER MODELS II: BEYOND ω-HUGE

We investigate large cardinal axioms beyond the level of ω-huge in context of the universality of the suitable extender models of [Suitable Extender Models I, J. Math. Log.10 (2010) 101–339]. We show that there is an analog of AD ℝ at the level of ω-huge, more precisely the construction of the minimum model of AD ℝ generalizes to the level of V λ+1 . This allows us to formulate the indicated generalization of AD ℝ and then to prove that if the axiom holds in V at a proper class of λ then in every suitable extender model, the axiom holds at a proper class of λ (provided the relevant ω-huge embeddings can be chosen to preserve the suitable extender model).