# Universal indestructibility

• A. W.~Apter and J. D. Hamkins, “Universal indestructibility,” Kobe J.~Math, vol. 16, iss. 2, pp. 119-130, 1999.
@article {ApterHamkins99:UniversalIndestructibility,
AUTHOR = {Arthur W.~Apter and Joel David Hamkins},
TITLE = {Universal indestructibility},
JOURNAL = {Kobe J.~Math},
FJOURNAL = {Kobe Journal of Mathematics},
VOLUME = {16},
YEAR = {1999},
NUMBER = {2},
PAGES = {119--130},
ISSN = {0289-9051},
MRCLASS = {03E55 (03E35)},
MRNUMBER = {1745027 (2001k:03112)},
MRNUMBER = {1 745 027},
eprint = {math/9808004},
archivePrefix = {arXiv},
primaryClass = {math.LO},
url = {http://wp.me/p5M0LV-12},
}

From a suitable large cardinal hypothesis, we provide a model with a supercompact cardinal in which universal indestructibility holds: every supercompact and partially supercompact cardinal kappa is fully indestructible by kappa-directed closed forcing. Such a state of affairs is impossible with two supercompact cardinals or even with a cardinal which is supercompact beyond a measurable cardinal.