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

• J. D. Hamkins, “Extensions with the approximation and cover properties have no new large cardinals,” Fund.~Math., vol. 180, iss. 3, p. 257–277, 2003.
[Bibtex]
@article{Hamkins2003:ExtensionsWithApproximationAndCoverProperties,
AUTHOR = {Hamkins, Joel David},
TITLE = {Extensions with the approximation and cover properties have no new large cardinals},
JOURNAL = {Fund.~Math.},
FJOURNAL = {Fundamenta Mathematicae},
VOLUME = {180},
YEAR = {2003},
NUMBER = {3},
PAGES = {257--277},
ISSN = {0016-2736},
MRCLASS = {03E55 (03E40)},
MRNUMBER = {2063629 (2005m:03100)},
DOI = {10.4064/fm180-3-4},
URL = {http://wp.me/p5M0LV-2B},
eprint = {math/0307229},
archivePrefix = {arXiv},
primaryClass = {math.LO},
file = F,
}

If an extension $\bar V$ of $V$ satisfies the $\delta$-approximation and cover properties for classes and $V$ is a class in $\bar V$, then every suitably closed embedding $j:\bar V\to \bar N$ in $\bar V$ with critical point above $\delta$ restricts to an embedding $j\upharpoonright V:V\to N$ amenable to the ground model $V$. In such extensions, therefore, there are no new large cardinals above delta. This result extends work in my article on gap forcing.