Determinacy and the sharp function on objects of type K

Journal of Symbolic Logic 60 (4):1025-1053 (1995)
We 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, (Π 0 k ) * and (Π 0 k ) * + , which lie strictly between $\bigcup_{\beta and Δ(ω 2 -Π 1 1 ). Let ♯ k be the (partial) sharp function on objects of type k. We show that the determinancy of (Π 0 k ) * follows from $L \lbrack\ sharp_k \rbrack \models "\text{every object of type} k \text{has a sharp},$ and we show that the existence of indiscernibles for L[ ♯ k ] is equivalent to a slightly stronger determinacy hypothesis, the determinacy of (Π 0 k ) * +
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DOI 10.2307/2275873
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Donald A. Martin (1990). An Extension of Borel Determinacy. Annals of Pure and Applied Logic 49 (3):279-293.

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