具有全集的度量集合论
A metric set theory with a universal set
摘要：
受度量结构模型论思想的启发，我们引入了一个度量集合论，$\mathsf{MSE}$，它以有界量化为本原，由一个自然的度量可拓性公理组成（两个集合之间的距离是它们的扩展之间的Hausdorff距离）和一个近似的，非确定性的完全理解形式（对于任何实值公式$φ(x,y)$、参数$a$和$r < s$的元组，存在包含类$\{x: φ(x,a) \leq r\}$和类$\{x:φ(x,a) < s\}$的集合）。我们表明，$\mathsf{MSE}$是足以发展古典数学后，增加了适当的无穷公理。然后，我们构造规范的代表良好的序型，并证明超度量模型的$\mathsf{MSE}$总是包含外部病态序数，这是真实的所有模型。为了建立几个独立的结果，特别是一致性，我们构建了各种模型，包括伪有限模型和模型包含任意大的标准序数。最后，我们讨论了如何形式化$\mathsf{MSE}$在连续逻辑或Eukasiewicz逻辑。
英文摘要：
Motivated by ideas from the model theory of metric structures, we introduce a metric set theory, $\mathsf{MSE}$, which takes bounded quantification as primitive and consists of a natural metric extensionality axiom (the distance between two sets is the Hausdorff distance between their extensions) and an approximate, non-deterministic form of full comprehension (for any real-valued formula $φ(x,y)$, tuple of parameters $a$, and $r < s$, there is a set containing the class $\{x: φ(x,a) \leq r\}$ and contained in the class $\{x:φ(x,a) < s\}$). We show that $\mathsf{MSE}$ is sufficient to develop classical mathematics after the addition of an appropriate axiom of infinity. We then construct canonical representatives of well-order types and prove that ultrametric models of $\mathsf{MSE}$ always contain externally ill-founded ordinals, conjecturing that this is true of all models. To establish several independence results and, in particular, consistency, we construct a variety of models, including pseudo-finite models and models containing arbitrarily large standard ordinals. Finally, we discuss how to formalize $\mathsf{MSE}$ in either continuous logic or Łukasiewicz logic.
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