Mathematical Sciences: Subsystems of Second-Order Arithmetic

数学基础中的一个中心问题是：为了证明普通数学的具体定理，需要哪些集合存在公理？这里的“普通数学”指的是数学中不是公开设定的理论部分。例如，“普通数学”包括几何、微分方程、实数和复数分析、可数代数、可数组合学和完全可分度量空间的拓扑等学科。另一方面，《普通数学》不包括一般拓扑学和不可数基数算术等主题。研究发现，二阶算术语言是研究上述问题的合适语境。(这是因为“普通数学”在很大程度上可以用可数数学和可分数学来识别。)此外，许多普通数学的关键定理，当被形式化为二阶算术语句时，被证明为与证明它们所需的集合存在公理等价。(在弱基理论上证明了等价性。)令人惊讶的是，以这种方式产生的不等价集合存在公理的数量相当少。这是由辛普森和其他人开发的逆数学的主题，并将在当前项目下进一步发展。

A central question in the foundations of mathematics is: Which set existence axioms are needed in order to prove specific theorems of ordinary mathematics? Here "ordinary mathematics" refers to the parts of mathematics which are not overtly set theoretical. For instance "ordinary mathematics" includes subjects such as geometry, differential equations, real and complex analysis, countable algebra, countable combinatorics, and the topology of complete separable metric spaces. On the other hand, "ordinary mathematics" does not include topics such as general topology and uncountable cardinal arithmetic. It has been found that the language of second order arithmetic is an appropriate context for the study of the above question. (This is because "ordinary mathematics" is to a large extent identifiable with countable and separable mathematics.) Moreover, a great many key theorems of ordinary mathematics, when formalized as sentences of second order arithmetic, turn out to be equivalent to the set existence axioms which are needed to prove them. (The equivalent is proved over a weak base theory.) Surprisingly, the number of inequivalent set existence axioms which arise in this way is rather small. This is the theme of Reverse Mathematics, which has been developed by Simpson and others and will be further developed under the current project.