# Large cardinals with few measures

• J. Apter Arthur W.~and Cummings and J. D. Hamkins, “Large cardinals with few measures,” Proc.~Amer.~Math.~Soc., vol. 135, iss. 7, pp. 2291-2300, 2007.
@ARTICLE{ApterCummingsHamkins2006:LargeCardinalsWithFewMeasures,
AUTHOR = {Apter, Arthur W.~and Cummings, James and Hamkins, Joel David},
TITLE = {Large cardinals with few measures},
JOURNAL = {Proc.~Amer.~Math.~Soc.},
FJOURNAL = {Proceedings of the American Mathematical Society},
VOLUME = {135},
YEAR = {2007},
NUMBER = {7},
PAGES = {2291--2300},
ISSN = {0002-9939},
CODEN = {PAMYAR},
MRCLASS = {03E35 (03E55)},
MRNUMBER = {2299507 (2008b:03067)},
MRREVIEWER = {Tetsuya Ishiu},
DOI = {10.1090/S0002-9939-07-08786-2},
URL = {http://dx.doi.org/10.1090/S0002-9939-07-08786-2},
eprint = {math/0603260},
archivePrefix = {arXiv},
primaryClass = {math.LO},
file = F
}

We show, assuming the consistency of one measurable cardinal, that it is consistent for there to be exactly $\kappa^+$ many normal measures on the least measurable cardinal $\kappa$. This answers a question of Stewart Baldwin. The methods generalize to higher cardinals, showing that the number of $\lambda$-strong compactness or $\lambda$-supercompactness measures on $P_\kappa(\lambda)$ can be exactly $\lambda^+$, if $\lambda>\kappa$ is a regular cardinal. We conclude with a list of open questions. Our proofs use a critical observation due to James Cummings.