Stably Measurable Cardinals

P. D. Welch

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arXiv · 1901.05551

Stably Measurable Cardinals

P. D. Welch

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Abstract

We define a weak iterability notion that is sufficient for a number of arguments concerning $Σ_1$-definability at uncountable regular cardinals. In particular we give its exact consistency strength firstly in terms of the second uniform indiscernible for bounded subsets of $κ$: $u_2(κ)$, and secondly to give the consistency strength of a property of Lücke's. Theorem: The following are equiconsistent: (i) There exists $κ$ which is stably measurable; (ii) for some cardinal $κ$, $u_2(κ)=σ(κ)$; (iii) The {\boldmath $Σ_1$}-club property holds at a cardinal $κ$. Here $σ(κ)$ is the height of the smallest $M \prec_{Σ_1} H(κ^+)$ containing $κ+1$ and all of $H(κ)$.

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BibTeX RIS

P. D. Welch. 2019-01-16. Stably Measurable Cardinals. https://arxiv.org/abs/1901.05551

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