Determinacy and the sharp function on objects of type k

Abstract

AbstractWe characterize, in terms of determinacy, the existence of the least inner model of “every object of type k has a sharp.” For k ∈ ω, we define two classes of sets, and , which lie strictly between and . Let #k be the (partial) sharp function on objects of type k. We show that the determinacy of follows fromL[#k] ⊨ “every object of type k has a sharp”,and we show that the existence of indiscemibles for L[#k] is equivalent to a slightly stronger determinacy hypothesis, the determinacy of .