Journal of Symbolic Logic

Jonsson Cardinals, Erdos Cardinals, and the Core Model

W. J. Mitchell

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Abstract

We show that if there is no inner model with a Woodin cardinal and the Steel core model K exists, then every Jonsson cardinal is Ramsey in K, and every $\delta$-Jonsson cardinal is $\delta$-Erdos in K. In the absence of the Steel core model K we prove the same conclusion for any model L$[\mathscr{E}]$ such that either V = L$[\mathscr{E}]$ is the minimal model for a Woodin cardinal, or there is no inner model with a Woodin cardinal and V is a generic extension of L$[\mathscr{E}]$. The proof includes one lemma of independent interest: If V = L$[\mathscr{A}]$, where A $\subset$ $\kappa$ and $\kappa$ is regular, then L$_\kappa$[A] is a Jonsson algebra. The proof of this result, Lemma 2.5, is very short and entirely elementary.

Article information

Source
J. Symbolic Logic, Volume 64, Issue 3 (1999), 1065-1086.

Dates
First available in Project Euclid: 6 July 2007

Permanent link to this document
https://projecteuclid.org/euclid.jsl/1183745870

Mathematical Reviews number (MathSciNet)
MR1779751

Zentralblatt MATH identifier
0938.03076

JSTOR
links.jstor.org

Citation

Mitchell, W. J. Jonsson Cardinals, Erdos Cardinals, and the Core Model. J. Symbolic Logic 64 (1999), no. 3, 1065--1086. https://projecteuclid.org/euclid.jsl/1183745870


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