论文标题
基础性比较的逻辑没有选择的公理
The Logic of Cardinality Comparison Without the Axiom of Choice
论文作者
论文摘要
我们在Zermelo-Fraenkel设定理论的环境中工作，而无需假设选择公理。我们考虑使用布尔操作的设置以及比较基数的附加结构（从坎托里亚的注射意义上）。人们需要在布尔代数法律中添加哪些原则，不仅要推理集合的交叉点，联合和互补，还要推理集合的相对大小？我们给出完整的公理化。 一个特别有趣的案例是，当人们限制了Dedekind-Finite套装时。在这种情况下，一个人需要与关于不精确概率比较的理论完全相同的原则，中心原则是普遍的有限取消（包括特殊情况，包括逐$ m $）。在一般情况下，中央原则是阿基米德阶级中广义有限取消的限制版本，我们称之为涵盖的广义有限取消。
We work in the setting of Zermelo-Fraenkel set theory without assuming the Axiom of Choice. We consider sets with the Boolean operations together with the additional structure of comparing cardinality (in the Cantorian sense of injections). What principles does one need to add to the laws of Boolean algebra to reason not only about intersection, union, and complementation of sets, but also about the relative size of sets? We give a complete axiomatization. A particularly interesting case is when one restricts to the Dedekind-finite sets. In this case, one needs exactly the same principles as for reasoning about imprecise probability comparisons, the central principle being Generalized Finite Cancellation (which includes, as a special case, division-by-$m$). In the general case, the central principle is a restricted version of Generalized Finite Cancellation within Archimedean classes which we call Covered Generalized Finite Cancellation.
