Extensions with the approximation and cover properties have no new large cardinals

Joel David Hamkins¹

¹ The Graduate Center of The City University of New York Mathematics Program 365 Fifth Avenue New York, NY 10016, U.S.A.

Fundamenta Mathematicae, Tome 180 (2003) no. 3, pp. 257-277

Résumé

If an extension $V\subseteq{\overline{V}}$ satisfies the $\delta$ approximation and cover properties for classes and $V$ is a class in ${\overline{V}}$, then every suitably closed embedding $j:{\overline{V}}\to\overline{N}$ in ${\overline{V}}$ with critical point above $\delta$ restricts to an embedding $j{\upharpoonright} V$ amenable to the ground model $V$. In such extensions, therefore, there are no new large cardinals above $\delta$. This result extends work in \cite{Hamkins2001:GapForcing}.

Détail

DOI : 10.4064/fm180-3-4

Keywords: extension subseteq overline satisfies delta approximation cover properties classes class overline every suitably closed embedding overline overline overline critical point above delta restricts embedding upharpoonright amenable ground model nbsp extensions therefore there large cardinals above delta result extends work cite hamkins gapforcing