The asymmetry of mathematical determinacy: understanding why arithmetic is determinate and set theory indeterminate.

大多数数学家和哲学家含蓄地接受算术，即对自然数及其基本运算的数学研究，是确定的，也就是说，它的所有陈述要么是真的，要么是假的。尽管如此，他们中的许多人也怀疑集合论的某些陈述，数学中处理对象集合以及无穷大的部分，可能是不确定的，即缺乏真值。在我的项目中，我试图通过提供哲学和技术论证来为这些直觉奠定基础，其最终目标是支持一种形式的集合论多元主义，也就是说，与算术不同，集合论呈现了宇宙的多样性。在此过程中，我希望能够理解确定性的含义，更重要的是，我希望能够揭示我们日常使用数学的一些隐藏的特征，以及无限的一些奇迹。

Most mathematicians and philosophers implicitly accept that arithmetic, the mathematical investigation of natural numbers and their basic operations, is determinate, i.e. all of its statements are either true or false. Nonetheless, many of them also suspect that some statements of set theory, the part of mathematics that deals with collections of objects as well as infinity, could be indeterminate, i.e. lack a truth value. In my project, I try to ground these intuitions by providing philosophical and technical arguments whose ultimate goal is to support a form of set-theoretic pluralism, that is, the idea that, unlike arithmetic, set theory presents a multiplicity of universes. And, on the way there, I hope to grasp what it means for something to be determinate and, more importantly, to reveal some of the hidden features of our everyday use of mathematics and some of the wonders of the infinite.