An increasing amount of contemporary philosophy of mathematics posits, and theorizes in terms of special kinds of mathematical modality. The goal of this paper is to bring recent work on higher-order metaphysics to bear on the investigation of these modalities. The main focus of the paper will be views that posit mathematical contingency or indeterminacy about statements that concern the ‘width’ of the set theoretic universe, such as Cantor’s continuum hypothesis. Within a higher-order framework I show that contingency about the width of the set-theoretic universe refutes two orthodoxies concerning the structure of modal reality: the view that the broadest necessity has a logic of S5, and the ‘Leibniz biconditionals’ stating that what is possible, in the broadest sense of possible, is what is true in some possible world. Nonetheless, I suggest that the underlying picture of modal set-theory is coherent and has attractions.

越来越多的当代数学哲学根据特殊类型的数学模态进行假设和理论化。本文的目标是将高阶形而上学的最新研究成果用于对这些模式的研究。本文的主要焦点是对涉及集合论宇宙“宽度”的陈述提出数学偶然性或不确定性的观点，例如康托尔的连续统假设。在高阶框架内，我证明了关于集合论宇宙宽度的偶然性驳斥了关于模态现实结构的两种正统观点：最广泛的必然性具有 S5 逻辑的观点，以及“莱布尼兹双条件”指出什么从最广泛的意义上来说，“可能”是可能的世界中的真实情况。尽管如此，我认为模态集合论的基本图景是连贯的并且具有吸引力。