• J. D. Hamkins and W. Woodin, “Small forcing creates neither strong nor Woodin cardinals,” Proc.~Amer.~Math.~Soc., vol. 128, iss. 10, pp. 3025-3029, 2000.  
  Journal MR Citation arχiv

  @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.