Large cardinals with few measures

Abstract

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