概括原则与悖论研究的进展

On the Development of Comprehensive Principle and Paradoxes

悖论的出现与概括原则中的造集的任意性有很大关系。然而,历史上各种排除悖论的方案要么修改概括原则而过多地限制了概括原则的合理内容,要么修改与之配套的逻辑系统而导致失败。近年来,这些问题的研究又取得了新的进展,有必要历史地加以评介,本文从类型论、ZFC公理系统、BG公理系统、多值逻辑系统和中介数学系统五个方面综述逻辑数学悖论之解释方法的研究情况。

It is well known that the appearance of paradoxes has much to do with the arbitrariness of constructing a set in comprehension principle. However, various programmes in history which exclude paradoxes either excessively exclude the reasonable content of comprehension principle when revised, or change the logic system on which it is based, and so lead to failure. In recent years, the new achievement has been made on the research of these problems, so it is necessary to survey them historically. In this paper, wo outline the development of research in the explanatory methods of logical and mathematical paradoxes in five aspects, i.e.theory of types, ZFC axiomatic set theory, BG axiomatic set theory, Multiple-valued logic and medium mathematics.