Journal of Symbolic Logic

The Complexity of the Core Model

William J. Mitchell

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If there is no inner model with a cardinal $\kappa$ such that $o(\kappa) = \kappa^{++}$ then the set $K \cap H_{\omega_1}$ is definable over H$_{\omega_1}$ by a $\Delta_4$ formula, and the set $\{J_\alpha[\mathscr{U}] : \alpha < \omega_1\}$ of countable initial segments of the core model $K = L[\mathscr{U}]$ is definable over $H_{\omega_1}$ by a $\Pi_3$ formula. We show that if there is an inner model with infinitely many measurable cardinals then there is a model in which $\{J_\alpha [\mathscr{U}] : \alpha < \omega_1\}$ is not definable by any $\Sigma_3$ formula, and $K \cap H_{\omega_1}$ is not definable by any boolean combination of $\Sigma_3$ formulas.

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J. Symbolic Logic, Volume 63, Issue 4 (1998), 1393-1398.

First available in Project Euclid: 6 July 2007

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Mitchell, William J. The Complexity of the Core Model. J. Symbolic Logic 63 (1998), no. 4, 1393--1398.

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