Determinacy and the sharp function on objects of type K

Journal of Symbolic Logic 60 (4):1025-1053 (1995)
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Abstract

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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References found in this work

Set Theory.Thomas Jech - 1999 - Studia Logica 63 (2):300-300.
Descriptive Set Theory.Yiannis Nicholas Moschovakis - 1982 - Studia Logica 41 (4):429-430.
An extension of borel determinacy.Donald A. Martin - 1990 - Annals of Pure and Applied Logic 49 (3):279-293.
Determinacy and the sharp function on the reals.Derrick Albert DuBose - 1992 - Annals of Pure and Applied Logic 55 (3):237-263.

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