On unfoldable cardinals, ω-closed cardinals, and the beginning of the inner model hierarchy

Abstract.

Let κ be a cardinal, and let H _κ be the class of sets of hereditary cardinality less than κ ; let τ (κ) > κ be the height of the smallest transitive admissible set containing every element of {κ}∪H _κ . We show that a ZFC-definable notion of long unfoldability, a generalisation of weak compactness, implies in the core model K, that the mouse order restricted to H _κ is as long as τ. (It is known that some weak large cardinal property is necessary for the latter to hold.) In other terms we delimit its strength as follows: Theorem Con(ZFC+ω²-Π ¹ ₁-Determinacy) ⇒ ⇒Con(ZFC+V=K+∃ a long unfoldable cardinal ⇒ ⇒Con(ZFC+∀X(X ^# exists) + ‘‘$\forall D \subseteq \omega_1 D$ is universally Baire ⇔ ∃r∈R(D∈L(r)))’’, and this is set-generically absolute). We isolate a notion of ω-closed cardinal which is weaker than an ω₁-Erd\ os cardinal, and show that this bounds the first long unfoldable: Theorem Let κ be ω -closed. Then there is a long unfoldable ł<κ.