Extensions with the approximation and cover properties have no new large cardinals
Volume 180 / 2003
Fundamenta Mathematicae 180 (2003), 257-277
MSC: 03E55, 03E40.
DOI: 10.4064/fm180-3-4
Abstract
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}.
