反推数学与无穷

Reverse Mathematics and Infinity

无穷一直以来都是数学哲学中的一个基本问题,但不同的无穷观都未令人满意。反推数学这一数学新纲领给无穷研究带来了新视角。本文在简述数学中的无穷概念后,利用反推数学对无穷进行梳理。具体而言,首先,反推数学可将经典数学中的大部分无穷数学归约到有穷数学。其次,通过对超越二阶算术的高阶反推数学的研究,发现部分高阶数学在二阶数学中都有对应的部分。最后,初步探讨经典数学需要多大的无穷。

Infinity has always been a basic problem in the philosophy of mathematics,but different views of infinity are not satisfactory.Reverse mathematics,a new mathematics program,brings us a new method on infinity.After briefly introducing the concept of infinity in mathematics,this article uses reverse mathematics to sort out the concept.Specifically,first of all,reverse mathematics can finitely reduce most of the infinite mathematics in classical mathematics to a finite mathematics.Secondly,through the discussion of higher order reverse mathematics beyond the secondorder arithmetic,it is found that some parts of higherorder mathematics have corresponding parts in secondorder mathematics.Finally,preliminarily explore how big infinity is in classical mathematics.