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 (
- Award ID(s):
- 2047756
- PAR ID:
- 10413665
- Publisher / Repository:
- Springer Science + Business Media
- Date Published:
- Journal Name:
- Theory of Computing Systems
- Volume:
- 67
- Issue:
- 3
- ISSN:
- 1432-4350
- Format(s):
- Medium: X Size: p. 627-669
- Size(s):
- p. 627-669
- Sponsoring Org:
- National Science Foundation
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