# A conference in honor of W. Hugh Woodin’s 60th birthday, March 2015

I am pleased to announce the upcoming conference at Harvard celebrating the 60th birthday of W. Hugh Woodin.  See the conference web site for more information. Click on the image below for a large-format poster.

# The ground axiom is consistent with $V\ne{\rm HOD}$

• J. D. Hamkins, J. Reitz, and W. Woodin, “The ground axiom is consistent with $V\ne{\rm HOD}$,” Proc.~Amer.~Math.~Soc., vol. 136, iss. 8, pp. 2943-2949, 2008.
@ARTICLE{HamkinsReitzWoodin2008:TheGroundAxiomAndVequalsHOD,
AUTHOR = {Hamkins, Joel David and Reitz, Jonas and Woodin, W.~Hugh},
TITLE = {The ground axiom is consistent with {$V\ne{\rm HOD}$}},
JOURNAL = {Proc.~Amer.~Math.~Soc.},
FJOURNAL = {Proceedings of the American Mathematical Society},
VOLUME = {136},
YEAR = {2008},
NUMBER = {8},
PAGES = {2943--2949},
ISSN = {0002-9939},
CODEN = {PAMYAR},
MRCLASS = {03E35 (03E45 03E55)},
MRNUMBER = {2399062 (2009b:03137)},
MRREVIEWER = {P{\'e}ter Komj{\'a}th},
DOI = {10.1090/S0002-9939-08-09285-X},
URL = {http://dx.doi.org/10.1090/S0002-9939-08-09285-X},
file = F
}

Abstract. The Ground Axiom asserts that the universe is not a nontrivial set-forcing extension of any inner model. Despite the apparent second-order nature of this assertion, it is first-order expressible in set theory. The previously known models of the Ground Axiom all satisfy strong forms of $V=\text{HOD}$. In this article, we show that the Ground Axiom is relatively consistent with $V\neq\text{HOD}$. In fact, every model of ZFC has a class-forcing extension that is a model of $\text{ZFC}+\text{GA}+V\neq\text{HOD}$. The method accommodates large cardinals: every model of ZFC with a supercompact cardinal, for example, has a class-forcing extension with $\text{ZFC}+\text{GA}+V\neq\text{HOD}$ in which this supercompact cardinal is preserved.

# The necessary maximality principle for c.c.c. forcing is equiconsistent with a weakly compact cardinal

• W. Hamkins Joel D.~and Woodin, “The necessary maximality principle for c.c.c.\ forcing is equiconsistent with a weakly compact cardinal,” MLQ Math.~Log.~Q., vol. 51, iss. 5, pp. 493-498, 2005.
@ARTICLE{HamkinsWoodin2005:NMPccc,
AUTHOR = {Hamkins, Joel D.~and Woodin, W.~Hugh},
TITLE = {The necessary maximality principle for c.c.c.\ forcing is equiconsistent with a weakly compact cardinal},
JOURNAL = {MLQ Math.~Log.~Q.},
FJOURNAL = {MLQ.~Mathematical Logic Quarterly},
VOLUME = {51},
YEAR = {2005},
NUMBER = {5},
PAGES = {493--498},
ISSN = {0942-5616},
MRCLASS = {03E65 (03E55)},
MRNUMBER = {2163760 (2006f:03082)},
MRREVIEWER = {Tetsuya Ishiu},
DOI = {10.1002/malq.200410045},
URL = {http://dx.doi.org/10.1002/malq.200410045},
eprint = {math/0403165},
archivePrefix = {arXiv},
primaryClass = {math.LO},
file = F,
}

The Necessary Maximality Principle for c.c.c. forcing asserts that any statement about a real in a c.c.c. extension that could become true in a further c.c.c. extension and remain true in all subsequent c.c.c. extensions, is already true in the minimal extension containing the real. We show that this principle is equiconsistent with the existence of a weakly compact cardinal.

See related article on the Maximality Principle

# Small forcing creates neither strong nor Woodin cardinals

• 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.
@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://dx.doi.org/10.1090/S0002-9939-00-05347-8},
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.