Mathematical Sciences: Subsystems of Second-Order Arithmetic

数学基础中的一个中心问题是： 为了证明特定的集合存在公理， 普通数学的定理 这里的“普通数学” 指的是数学中没有被公开设定的部分 理论上的 例如，“普通数学”包括 几何、微分方程、真实的和 复分析，可数代数，可数组合学， 完备可分度量空间的拓扑。 另 一方面，“普通数学”不包括诸如 一般拓扑学和不可数基数算术。 已经发现，二阶语言 算术是研究上述问题的适当背景。 问题 (This是因为“普通数学”在很大程度上 可识别可数和可分离数学的范围。 此外，许多普通数学的关键定理，当 形式化为二阶算术的句子，结果是 等价于集合存在公理， 证明他们。 (The等价于弱基理论。） 令人惊讶的是，不等价集合存在公理的数量 这种方式产生的影响是相当小的。 这是一个主题 逆向数学，这是由辛普森和 并将在本项目下进一步发展。

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.