In the absence of the Axiom of Choice, the “small” cardinal ${\omega }_1$ can exhibit properties more usually associated with large cardinals, such as strong compactness and supercompactness. For a local version of strong compactness, we say that ${\omega }_1$ is X-strongly compact (where X is any set) if there is a fine, countably complete measure on ${\wp }_{{\omega }_1}(X)$. Working in $\mathsf{ZF}+\mathsf{DC}$, we prove that the $\wp ({\omega }_1)$-strong compactness and $\wp (\mathbb{R})$-strong compactness of ${\omega }_1$ are equiconsistent with $\mathsf{AD}$ and ${\mathsf{AD}}_{\mathbb{R}}+\mathsf{DC}$ respectively, where $\mathsf{AD}$ denotes the Axiom of Determinacy and ${\mathsf{AD}}_{\mathbb{R}}$ denotes the Axiom of Real Determinacy. The $\wp (\mathbb{R})$-supercompactness of ${\omega }_1$ is shown to be slightly stronger than ${\mathsf{AD}}_{\mathbb{R}}+\mathsf{DC}$, but its consistency strength is not computed precisely. An equiconsistency result at the level of ${\mathsf{AD}}_{\mathbb{R}}$ without $\mathsf{DC}$ is also obtained.

在没有选择公理的情况下，“小”主教 ${\omega }_{\mathrm{1个}}$可以表现出通常与大型红衣主教相关的特性，例如紧密性和超紧凑性。对于具有高度紧凑性的本地版本，我们说${\omega }_{\mathrm{1个}}$是X -strongly紧凑（其中X是任何集）如果有一个细，可数完整度量上${\wp }_{{\omega }_{\mathrm{1个}}}（X）$。在工作$\mathsf{采埃孚}+\mathsf{直流电}$，我们证明 $\wp （{\omega }_{\mathrm{1个}}）$紧密度高 $\wp （\mathbb{[R}）$的紧凑性 ${\omega }_{\mathrm{1个}}$ 与...一致 $\mathsf{广告}$ 和 ${\mathsf{广告}}_{\mathbb{[R}}+\mathsf{直流电}$ 分别在哪里 $\mathsf{广告}$ 表示确定性公理，并且 ${\mathsf{广告}}_{\mathbb{[R}}$表示实际确定性公理。的$\wp （\mathbb{[R}）$的超紧凑 ${\omega }_{\mathrm{1个}}$ 被证明比 ${\mathsf{广告}}_{\mathbb{[R}}+\mathsf{直流电}$，但是其一致性强度无法精确计算。在以下级别的等一致性结果${\mathsf{广告}}_{\mathbb{[R}}$ 没有 $\mathsf{直流电}$ 也获得了。