Journal of Symbolic Logic

Determinacy and the Sharp Function on Objects of Type k

Derrick Albert Dubose

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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 $k \in \omega$, 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 $(\Pi^0_k)^\ast_+$.

Article information

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

Dates
First available in Project Euclid: 6 July 2007

Permanent link to this document
https://projecteuclid.org/euclid.jsl/1183744862

Mathematical Reviews number (MathSciNet)
MR1367195

Zentralblatt MATH identifier
0843.03030

JSTOR
links.jstor.org

Citation

Dubose, Derrick Albert. Determinacy and the Sharp Function on Objects of Type k. J. Symbolic Logic 60 (1995), no. 4, 1025--1053. https://projecteuclid.org/euclid.jsl/1183744862


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