克里普克-普拉特集合论的一个轻微弱化

中文摘要 AI 辅助

弱集合论$\boldsymbol{\textsf{ReR}}$是通过将有界收集方案替换为有界替换方案，从克里普克-普拉特集合论（$\boldsymbol{\textsf{KP}}$）得到的。我们证明$\boldsymbol{\textsf{ReR}}$可证$\boldsymbol{\textsf{TCo}}$，该命题断言每个集合都包含于一个传递集合中。这一结论用于说明，在$\boldsymbol{\textsf{ReR}}$和$\boldsymbol{\textsf{KP}}$中加入无穷公理的否定后得到的理论具有相同的后承。我们对$\boldsymbol{\textsf{TCo}}$的证明依赖于$\boldsymbol{\textsf{ReR}}$中类基础片段的可用性。为了证明这种依赖的必要性，即使在无穷存在的情况下，我们构建了一个包含有界分离、有界收集、无穷公理、幂集公理、正则公理和选择公理的$\boldsymbol{\textsf{ZF}}$的重要片段的模型，其中$\boldsymbol{\textsf{TCo}}$不成立。

英文摘要

The weak set theory $\mathsf{ReR}$ is obtained from Kripke-Platek Set Theory ($\mathsf{KP}$) by replacing the bounded collection scheme with the bounded replacement scheme. We show that $\mathsf{ReR}$ proves $\mathsf{TCo}$, which asserts that every set is contained in a transitive set. This is used to show that the theories obtained by adding the negation of the axiom of infinity to $\mathsf{ReR}$ and $\mathsf{KP}$ have the same consequences. Our proof of $\mathsf{TCo}$ relies on the availability of a fragment of class foundation in $\mathsf{ReR}$. To demonstrate the necessity of this reliance, even in the presence of infinity, we build a model of a significant fragment of $\mathsf{ZF}$ that includes bounded separation and collection, infinity, powerset, regularity and the axiom of choice, in which $\mathsf{TCo}$ fails.

发表机构

• University of Chester（切斯特大学）

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