Woodin cardinals, Shelah cardinals, and the Mitchell-Steel core model

Abstract

Theorem 4 is a characterization of Woodin cardinals in terms of Skolem hulls and Mostowski collapses. We define weakly hyper-Woodin cardinals and hyper-Woodin cardinals. Theorem 5 is a covering theorem for the Mitchell-Steel core model, which is constructed using total background ex-tenders. Roughly, Theorem 5 states that this core model correctly computes successors of hyper-Woodin cardinals. Within the large cardinal hierarchy, in increasing order we have: measurable Woodin, weakly hyper-Woodin, Shelah, hyper-Woodin, and superstrong cardinals. (The comparison of Shelah versus hyper-Woodin is due to James Cummings.) We begin by recalling the definition of a Woodin cardinal. If κ < λ are cardinals and S is a set, then κ is λ-S-strong iff there is a transitive class N and an elementary embedding j : V −→ N with κ = crit(j) , j(κ) ≥ λ and j(S) ∩ H λ = S ∩ H λ. If κ < δ, then κ is < δ-S-strong iff κ is λ-S-strong for every λ < δ. Finally, δ is a Woodin cardinal iff δ is a strongly inaccessible cardinal and for every S ⊆ H δ , there exists κ < δ such that κ is < δ-S-strong. It is well-known that the existence of an embedding that witnesses that κ is λ-S-strong is equivalent to the existence of an extender E ⊂ H λ which gives rise to such an embedding through an ultrapower construction. Thus δ is a Woodin cardinal is a Π 1 1 property of H δ. We will see how to express δ is a Woodin cardinal in terms of the Mostowski collapses of Skolem hulls. The uniformity with which this can be done is somewhat surprising. In the second part of this paper, we apply this characterization to core model theory. For each cardinal θ, let θ be a wellordering of H θ. Whenever we refer to H θ as a structure, we mean the structure H θ , ∈ , , θ. Suppose that M is transitive and π : M −→ H θ is an elementary embedding. Note that, in this case, π is the inverse of the Mostowski collapse isomorphism for the range of π. Let κ = crit(π)