- A. W.~Apter, J. 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 = {Arthur W.~Apter and James Cummings and Joel David Hamkins}, 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://jdh.hamkins.org/largecardinalswithfewmeasures/}, 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.