- J. D. Hamkins and W. Woodin, “Small forcing creates neither strong nor Woodin cardinals,” Proc.~Amer.~Math.~Soc., vol. 128, iss. 10, p. 3025–3029, 2000.

[Bibtex]`@article {HamkinsWoodin2000:SmallForcing, AUTHOR = {Hamkins, Joel David and Woodin, W.~Hugh}, TITLE = {Small forcing creates neither strong nor {W}oodin cardinals}, JOURNAL = {Proc.~Amer.~Math.~Soc.}, FJOURNAL = {Proceedings of the American Mathematical Society}, VOLUME = {128}, YEAR = {2000}, NUMBER = {10}, PAGES = {3025--3029}, ISSN = {0002-9939}, CODEN = {PAMYAR}, MRCLASS = {03E35 (03E55)}, MRNUMBER = {1664390 (2000m:03121)}, MRREVIEWER = {Carlos A.~Di Prisco}, DOI = {10.1090/S0002-9939-00-05347-8}, URL = {http://jdh.hamkins.org/smallforcing-w/}, eprint = {math/9808124}, archivePrefix = {arXiv}, primaryClass = {math.LO}, }`

After small forcing, almost every strongness embedding is the lift of a strongness embedding in the ground model. Consequently, small forcing creates neither strong nor Woodin cardinals.