AI 中文总结
针对独立于ZFC的怀特海问题，研究者在浓缩数学中给出类似问题的肯定解答，并证明大小为κ的阿贝尔群A是自由的等价于特定Ext函子取值为0。
AI 中文摘要
怀特海问题询问，满足$\text{Ext}^1(A,\boldsymbol{\text{Z}})=0$的每个阿贝尔群$A$是否都是自由的，该问题独立于ZFC公理系统。不过，该问题在浓缩数学中有一个类似问题，可得到肯定回答。在本注记中，我们给出新证明：大小为$\boldsymbol{\text{κ}}$的阿贝尔群$A$是自由的，当且仅当$\boldsymbol{\text{Ext}}^1(\boldsymbol{\text{A}},\boldsymbol{\text{Z}})(S_\boldsymbol{\text{κ}})=0$，其中$S_\boldsymbol{\text{κ}}$是$\text{Add}(\boldsymbol{\text{ω}},\boldsymbol{\text{κ}})$的布尔完备化的斯通空间。
英文摘要
The Whitehead problem, which asks whether every abelian group $A$ satisfying $\mathrm{Ext}^1(A,\mathbb{Z}) = 0$ is free, is independent of ZFC. However, this problem has an analogue in condensed mathematics that can be answered affirmatively. In this note, we give a new proof that an abelian group $ A $ of size $κ$ is free if and only if $\underline{\mathrm{Ext}}^1(\underline{A}, \underline{\mathbb{Z}})(S_κ) = 0$, where $ S_κ$ is the Stone space of the Boolean completion of $\operatorname{Add}(ω,κ)$.
