Partial realizations of Hilbert's program
Partial realizations of Hilbert's program
复制标题
希尔伯特纲领的部分实现
DOI：
10.1017/s0022481200028309
复制
发表时间：
1988
影响因子：
0.6
通讯作者：
S. G. Simpson
中科院分区：
文献类型：
--
作者：
S. G. Simpson
§0.导论.以下是我对“希尔伯特纲领60年后”研讨会的贡献，该研讨会由美国哲学协会和符号逻辑协会联合主办。该研讨会于1985年12月29日在华盛顿举行。C.小组成员是所罗门费弗曼、达格·普拉维茨和我自己。主持人是Wilfried Sieg。我在这里讨论的研究部分得到了NSF Grant DMS-8317874的支持。我感谢这次及时的研讨会的组织者，这次研讨会是关于一个重要的主题。作为一名数学家，我特别重视有机会向主要由哲学家组成的听众发表演讲。这是事实，我被要求集中在数学方面的希尔伯特的计划。但是，由于希尔伯特的纲领只涉及数学的基础，对数学方面的限制实际上根本没有限制。希尔伯特赋予一种特殊的角色，以某种有限的数学推理。希尔伯特纲领的实质是通过有限论的还原来证明所有的集合论数学。现在大家都知道，这项任务是无法完成的。任何这样的可能性都被哥德尔定理所驳斥。然而，最近的研究已经揭示了希尔伯特计划的重要部分实现的可行性。尽管哥德尔定理，人们可以给一个有限约化的大部分无穷数学，包括许多最著名的非建设性定理。我在这里的目的是提请注意这些现代发展。
§0. Introduction. What follows is a write-up of my contribution to the symposium “Hilbert's Program Sixty Years Later” which was sponsored jointly by the American Philosophical Association and the Association for Symbolic Logic. The symposium was held on December 29,1985 in Washington, D. C. The panelists were Solomon Feferman, Dag Prawitz and myself. The moderator was Wilfried Sieg. The research which I discuss here was partially supported by NSF Grant DMS-8317874. I am grateful to the organizers of this timely symposium on an important topic. As a mathematician I particularly value the opportunity to address an audience consisting largely of philosophers. It is true that I was asked to concentrate on the mathematical aspects of Hilbert's program. But since Hilbert's program is concerned solely with the foundations of mathematics, the restriction to mathematical aspects is really no restriction at all. Hilbert assigned a special role to a certain restricted kind of mathematical reasoning known as finitistic. The essence of Hilbert's program was to justify all of set-theoretical mathematics by means of a reduction to finitism. It is now well known that this task cannot be carried out. Any such possibility is refuted by Gödel's theorem. Nevertheless, recent research has revealed the feasibility of a significant partial realization of Hilbert's program. Despite Gödel's theorem, one can give a finitistic reduction for a substantial portion of infinitistic mathematics including many of the best-known nonconstructive theorems. My purpose here is to call attention to these modern developments.