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- W1504378551 abstract "In recent years, the Axiom of Determinateness (AD) has yielded numerous results concerning the size and properties of the first ω-many uncountable cardinals. Briefly, these results began with Solovay's discovery that ℵ 1 and ℵ 2 are measurable [8], [3], continued with theorems of Solovay, Martin, and Kunen concerning infinite-exponent partition relations [6], [3], Martin's proof that ℵ n has confinality ℵ 2 for 1 < n < ω, and very recently, Kleinberg's proof that the ℵ n are Jonsson cardinals [4]. This paper was inspired by a very recent result of Martin from AD that ℵ 1 is ℵ 2 -super compact. It was known for some time that AD implies ℵ 1 is α-strongly compact for all ℵ < θ (where θ is the least cardinal onto which 2 ω cannot be mapped, quite a large cardinal under AD), and that AD R implies that ℵ 1 , is α-super compact for all α < θ. A key open question had been whether or not ℵ 1 is super compact under AD alone. This paper comments on the method of Martin in several different ways. In §2, we will prove that ℵ 1 is ℵ 2 -super compact, and then generalize the method to show that ℵ 2 is ℵ 3 -strongly compact. In addition, we will demonstrate a limitation in the method by showing that the possible measures obtained on are not normal, and that the method cannot be extended to show that ℵ 2 is ℵ 4 -strongly compact." @default.
- W1504378551 created "2016-06-24" @default.
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- W1504378551 date "1978-09-01" @default.
- W1504378551 modified "2023-09-26" @default.
- W1504378551 title "On the compactness of ℵ1 and ℵ2" @default.
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- W1504378551 doi "https://doi.org/10.2307/2273517" @default.
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