Polynomial-Time Axioms of Choice and Polynomial-Time Cardinality

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