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

On unfoldable cardinals, !-closed cardinals, and the beginning of the inner model hierarchy Abstract. Let be a cardinal, and let H be the class of sets of hereditary cardinality less than ; let ( ) > be the height of the smallest transitive admissible set containing every element of H . 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 is as long as . (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 + !21 1 -Determinacy) ) ) Con(ZFC + V = K + 9 a long unfoldable cardinal ) ) Con(ZFC + 8C(X# exists) + “8D !1 (D is universally Baire , 9r 2 R(D 2 L(r)))”, and this is set-generically absolute). We isolate a notion of !-closed cardinal which is weaker than an !1-Erdős cardinal, and show that this bounds the first long unfoldable: Theorem Let be !-closed. Then there is a long unfoldable < .