Journal of Symbolic Logic

Determinacy and the Sharp Function on Objects of Type k

Derrick Albert Dubose

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We characterize, in terms of determinacy, the existence of the least inner model of "every object of type $k$ has a sharp." For $k \in \omega$, we define two classes of sets, $(\Pi^0_k)^\ast$ and $(\Pi^0_k)^\ast_+$, which lie strictly between $\bigcup_{\beta < \omega^2} (\beta-\Pi^1_1)$ and $\Delta(\omega^2-\Pi^1_1)$. Let $\sharp_k$ be the (partial) sharp function on objects of type $k$. We show that the determinancy of $(\Pi^0_k)^\ast$ 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 $L\lbrack \sharp_k \rbrack$ is equivalent to a slightly stronger determinacy hypothesis, the determinacy of $(\Pi^0_k)^\ast_+$.

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J. Symbolic Logic, Volume 60, Issue 4 (1995), 1025-1053.

First available in Project Euclid: 6 July 2007

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Dubose, Derrick Albert. Determinacy and the Sharp Function on Objects of Type k. J. Symbolic Logic 60 (1995), no. 4, 1025--1053.

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