集合论多宇宙观述评

On the Set Theory Multiverse View

集合论作为数学基础的成功使得关于数学对象的本体论问题被归约为关于集合的本体论问题。传统的(集合)实在论强调集合的存在,从而人们可以谈论关于这些集合的概念是否符合那个客观事实。而作者将在本文中介绍并评析另一种逐渐流行起来的集合论哲学——多宇宙观(Multiverse View)。这一立场强调存在许多集合论宇宙以及相应的集合概念。在该语境下,传统实在论则相应地被看作是一种唯一宇宙观(Universe View)的立场。两者的争论是当前数学基础领域的热门话题。

Set theory has been widely accepted as the foundation of mathematics. A philosophical significance of the fact is that the ontology about mathematical objects is reduced to the questions about sets. Realism （on sets） in the traditional sense insists on the existence of sets so that it is legitimate to talk about if a concept of sets coincides with the fact of sets. The author will introduce and also make comments on another rising opinion on the philosophy of set theory, namely the Muhiverse View. This new position emphasizes the existence of multiple set theory universes and many coordinate concepts of sets. In this context, realism in the traditional sense is regarded as an universe view, i.e. there is only one absolute universe of sets. The competition of the two opinions is a major topic about the foundation of mathematics.