Abstract:We propose that failures of the axiom of choice, that is, surjective functions admitting no sections, can be reasonably classified by means of invariants borrowed from algebraic topology. We show that cohomology, when defined so that its usual exactness properties hold even in the absence of the axiom of choice, is adequate for detecting failures of this axiom in the following sense. If a set $X$, viewed as a discrete space, has trivial first cohomology for all coefficient groups, then every $X$-indexed family of nonempty sets has a choice function. We also obtain related results when the coefficient groups are required to be abelian or well-orderable. In particular, we show that, if all discrete spaces have trivial first cohomology for all abelian coefficient groups, then the axiom of choice holds.
R. Diaconescu, Non-abelian cohomology à la Giraud, Kategorien, Tagungsbericht, Oberwolfach, June-July 1975.
R. J. Gauntt, Some restricted versions of the axiom of choice, Notices Amer. Math. Soc. 15 (1968), 351.
- Jean Giraud, Cohomologie non abélienne, Die Grundlehren der mathematischen Wissenschaften, Band 179, Springer-Verlag, Berlin-New York, 1971 (French). MR 0344253
- Paul E. Howard, Limitations on the Fraenkel-Mostowski method of independence proofs, J. Symbolic Logic 38 (1973), 416–422. MR 381989, DOI 10.2307/2273037
- Thomas J. Jech, The axiom of choice, Studies in Logic and the Foundations of Mathematics, Vol. 75, North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., Inc., New York, 1973. MR 0396271
- Andrzej Mostowski, Axiom of choice for finite sets, Fund. Math. 33 (1945), 137–168. MR 16352, DOI 10.4064/fm-33-1-137-168
- John Truss, Finite axioms of choice, Ann. Math. Logic 6 (1973/74), 147–176. MR 347603, DOI 10.1016/0003-4843(73)90007-7
- © Copyright 1983 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 279 (1983), 257-269
- MSC: Primary 03E25; Secondary 03G30, 55N25, 55N99
- DOI: https://doi.org/10.1090/S0002-9947-1983-0704615-7
- MathSciNet review: 704615