James Cummings | Joel David Hamkins

[bibtex key=ApterCummingsHamkins2013:SingularCardinalsAndStrongExtenders]

Brent Cody asked the question whether the situation can arise that one has an elementary embedding $𝑗 : 𝑉 \to 𝑀$ witnessing the $𝜃$ -strongness of a cardinal $𝜅$ , but where $𝜃$ is regular in $𝑀$ and singular in $𝑉$ .

In this article, we investigate the various circumstances in which this does and does not happen, the circumstances under which there exist a singular cardinal $𝜇$ and a short $(𝜅, 𝜇)$ -extender $𝐸$ witnessing “ $𝜅$ is $𝜇$ -strong”, such that $𝜇$ is singular in $𝑈 𝑙 𝑡 (𝑉, 𝐸)$ .

[bibtex key=ApterCummingsHamkins2006:LargeCardinalsWithFewMeasures]

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

Joel David Hamkins

mathematics and philosophy of the infinite

Tag Archives: James Cummings

Singular cardinals and strong extenders

Large cardinals with few measures

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