We introduce the notion of a critical cardinal as the critical point of sufficiently strong elementary embedding between transitive sets. Assuming the Axiom of Choice this is equivalent to measurability, but it is wellknown that Choice is necessary for the equivalence. Oddly enough, this central notion was never investigated on its own before. We prove a technical criterion for lifting elementary embeddings to symmetric extensions, and we use this to show that it is consistent relative to a supercompact cardinal that there is a critical cardinal whose successor is singular.

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关键红雀

我们引入了临界基数的概念作为传递集之间足够强的基本嵌入的临界点。假设选择公理，这等价于可测量性，但众所周知，选择是等价的必要条件。奇怪的是，这个中心概念以前从未被单独研究过。我们证明了将基本嵌入提升到对称扩展的技术标准，并且我们用它来证明相对于超紧基数，存在一个其后继是奇异的临界基数是一致的。