We improve on results and constructions by Apter, Dimitriou, Gitik, Hayut, Karagila, and Koepke concerning large cardinals, ultrafilters, and cofinalities without the axiom of choice. In particular, we show the consistency of the following statements from certain assumptions: the first supercompact cardinal can be the first uncountable regular cardinal, all successors of regular cardinals are Ramsey, every sequence of stationary sets in ${\aleph }_n$ is mutually stationary, an infinitary Chang conjecture holds for the cardinals ${\aleph }_{2n}$, and all ${\aleph }_n$ are singular. In each of the cases, our results either weaken the hypotheses or strengthen the conclusions of known proofs.

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关于Gitik模型的评论以及Lévy崩溃产品的对称扩展

我们改进了Apter，Dimitriou，Gitik，Hayut，Karagila和Koepke的结果和结构，这些结构和结构涉及大型主教，超滤器和最终选择，而没有选择公理。特别地，我们从某些假设中证明以下陈述的一致性：第一个超紧凑基数可以是第一个不可数的常规基数，所有常规基数的后继都是Ramsey，固定序列的每个序列在${\aleph }_{ñ}$ 是相互固定的，一个无限长张猜想适用于红衣主教 ${\aleph }_{2ñ}$， 和所有 ${\aleph }_{ñ}$是单数。在每种情况下，我们的结果要么削弱假设，要么加强已知证据的结论。