通向实数的第三条路——《算术的基本规律》中的实数理论

The Third Approach to Real Numbers:The Theory of Real Numbers in Basic Laws of Arithmetic

弗雷格的逻辑主义的一个组成部分是将实数理论还原为逻辑。在《算术的基本规律》中,实数被定义为量域中的量的比例,而量域是一个属于正类的类。尽管《算术的基本规律》中的系统有矛盾,但是这本著作中的实数理论能以一致的方式加以重构。库契拉选择在集合论的框架中重构它。他证明了弗雷格的量域和实数集是稠密连续有序且具有阿基米德性的阿贝尔群。本文在库契拉的重构的基础上,进一步指出它们是戴德金连续的阿基米德有序域。

One part of Frege’s logicism is to reduce the theory of real numbers to logic.In Basic Laws of Arithmetic,real numbers are intended to be defined as the ratios of quantities in a domain of quantities,which is a class belonging to a positive class.Although the system established in the work is inconsistent,the theory of real numbers presentend therein can be reconstructed in a consistent way.Kutschera chose to reconstruct it in the set-theoretic setting and proved that both a Fregean domain of quantities and the set of Fregean real numbers are dense,continuous and ordered abelian groups with Archimedean property.On the basis of his work,it is further shown in the paper that they are also Dedekind-continuous,Archimedean ordered fields.