- 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.

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