There is no single canonical polynomial-time version of the Axiom of Choice (AC); several statements of AC that are equivalent in Zermelo-Fraenkel (ZF) set theory are already inequivalent from a constructive point of view, and are similarly inequivalent from a complexity-theoretic point of view. In this paper we show that many classical formulations of AC, when restricted to polynomial time in natural ways, are equivalent to standard complexity-theoretic hypotheses, including several that were of interest to Selman. This provides a unified view of these hypotheses, and we hope provides additional motivation for studying some of the lesser-known hypotheses that appear here. Additionally, because several classical forms of AC are formulated in terms of cardinals, we develop a theory of polynomial-time cardinality. Nerode & Remmel (Contemp. Math. 106, 1990 and Springer Lec. Notes Math. 1432, 1990) developed a related theory, but restricted to unary sets. Downey (Math. Reviews MR1071525) suggested that such a theory over larger alphabets could have interesting connections to more standard complexity questions, and we illustrate some of those connections here. The connections between AC, cardinality, and complexity questions also allow us to highlight some of Selman’s work. We hope this paper is more of a beginning than an end, introducing new concepts and raising many new questions, ripe for further research.

选择公理 (AC) 没有单一的规范多项式时间版本；Zermelo-Fraenkel (ZF) 集合论中等价的 AC 的几个陈述从构造性的观点来看已经是不等价的，并且从复杂性理论的观点来看同样是不等价的。在本文中，我们展示了 AC 的许多经典公式，当以自然方式限制在多项式时间时，等同于标准的复杂性理论假设，包括 Selman 感兴趣的几个。这提供了这些假设的统一观点，我们希望为研究此处出现的一些鲜为人知的假设提供额外的动力。此外，由于 AC 的几种经典形式是根据基数制定的，因此我们开发了多项式时间基数理论。尼罗德和雷梅尔 (当代。数学。106, 1990 和 Springer Lec。笔记数学。1432, 1990) 发展了相关理论，但仅限于一元集。唐尼 (Math. Reviews MR1​​071525) 建议，这种关于更大字母表的理论可能与更标准的复杂性问题有有趣的联系，我们在这里说明了其中的一些联系。AC、基数和复杂性问题之间的联系也使我们能够突出 Selman 的一些工作。我们希望这篇论文更多的是一个开始而不是结束，引入新概念并提出许多新问题，为进一步研究做好准备。