Large cardinals with few measures

Abstract

Abstract. 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.

Coding into HOD via normal measures with some applications

“…We will take this opportunity to discuss a generalization of Hamkins’ Gap Forcing Theorem 6, 8 (as it is stated in 1), as its results are used extensively throughout this paper. A forcing notion \documentclass{article}\usepackage{amssymb}\pagestyle{empty}\begin{document}${\mathbb {P}}$\end{document} (and the forcing extension to which it gives rise) admits a closure point at δ if it factors as \documentclass{article}\usepackage{amssymb}\pagestyle{empty}\begin{document}${\mathbb Q}\,{*}\,\dot{{\mathbb R}}$\end{document}, where \documentclass{article}\usepackage{amssymb}\pagestyle{empty}\begin{document}${\mathbb Q}$\end{document} is nontrivial, \documentclass{article}\usepackage{amssymb}\pagestyle{empty}\begin{document}$|{\mathbb Q}| \le \delta$\end{document}, and \documentclass{article}\usepackage{amssymb}\pagestyle{empty}\begin{document}$\Vdash _{\mathbb Q}``\dot{{\mathbb R}}$\end{document} is δ‐strategically closed.” Our arguments will rely on the following consequence of the main result of 10.…”

Level by level equivalence and the number of normal measures over Pκ(λ)

“…To do this, we use an argument due to Cummings, which also appears in the proof of the Main Theorem of [4] and the proof of Lemma 2.1 of [2]. First, note that by our assumptions on V , V "κ carries exactly κ ++ = 2 2 κ many normal measures".…”

“…Aside from the classical result (see [11]) that if κ is 2 [λ] <κ supercompact, then P κ (λ) carries exactly 2 2 [λ] <κ many κ-additive, fine, normal measures (the maximal number), and the more recent results of [4] that when λ ≥ κ is regular, it is consistent relative to the appropriate assumptions for κ to be λ supercompact and for P κ (λ) to carry fewer than the maximal number of κ-additive, fine, normal measures, not much has been known concerning models for supercompactness and the number of normal measures P κ (λ) can carry.…”

“…For anything left unexplained, readers are urged to consult [4]. When forcing, q ≥ p means that q is stronger than p. For κ a regular cardinal and λ ≥ κ any cardinal, Add(κ, 1) is the standard partial ordering for adding a single Cohen subset of κ, and Coll(κ, λ) is the standard collapse partial ordering (originally used by Cohen) for collapsing λ to κ.…”

How many normal measures can \alephω+ 1carry?

“…To do this, we use a new method due to James Cummings, which appears in [5] in a broader context. We isolate Cummings' techniques in the following lemma, which we state in a slightly generalized form.…”