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

Abstract

Let κ be a cardinal, and let Hκ 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}∪Hκ. 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 Hκ 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: TheoremCon(ZFC+ω2-Π 11-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 l<κ.