希尔伯特的有穷数学

Hilbert's Finite Mathematics

希尔伯特为了一劳永逸地解决数学基础问题,提出了著名的希尔伯特纲领。该纲领旨在把数学归约到毋庸置疑的有穷数学。遗憾的是,希尔伯特本人并未对有穷数学给出具体形式化。在简介希尔伯特有穷数学的基本思想后,梳理了各种不同的形式化系统:初始递归算术(PRA)、ZFC的有穷数学系统(Fin(ZFC))和基本算术(EA),并对PRA是希尔伯特的有穷数学进行辩护和简要述评。

To solve the problem of the foundation of mathematics once and for all, Hilbert has put forward the famous Hilbert＇s program. The program aims to reduce mathematics to indubitable finite mathematics. Unfortunately, Hilbert himself did not give a specific formal definition of finite mathematics. In this paper, an introduction to the basic idea of Hilbert＇s finite mathematics is given and various formal systems of Hilbert＇s finite mathematics are sorted out： Primitive Recursive Arithmetic（PRA）, finite mathematics of ZFC（Fin（ZFC）） and Elementary Arithmetic（EA）. After defending that PRA is the best formal system of Hilbert＇s finite mathematics, brief comments are presented on PRA.