经典数学的逻辑基础

The Logical Basis of Classical Mathematics

经典数学理论的逻辑完全是一阶逻辑还是也需要二阶逻辑？逻辑学家对此一直有争议。这种争议大约开始于20世纪20年代，但是似乎直到现在还未尘埃落定。在普遍接受反基础主义的前提下，数学基础的研究任务不再是为数学的各个分支寻找最大程度上免于理性怀疑的基础，而是在重构数学分支的过程中给出各个数学分支间的关系，描绘出数学的大图景。在这样的背景下，

Whether first-order logic is all of logic of classical mathematics or not is controversial.In the spirit of anti-foundationalism,the task of study on foundation of mathematics is not to provide the foundation of mathematics which is immune to rational doubt maximally,but to find the relations of branches of mathematics and try to describe the whole picture of mathematics.In this background,the study of foundation of mathematics cannot exclude the roles of second-order logic.With the premise that there is no sharp border between logic and mathematics,second-order logic is also logic.And the epistemology of second order logic is more natural.