nLab
small violations of choice

Contents

Statement

The axiom of small violations of choice, or SVC, introduced by Andreas Blass (1979), asserts that

Note that if S=1S=1, then every set is a subquotient of a choice set; but choice sets are closed under subsets and quotients, so this means that every set is choice, i.e. that the axiom of choice holds. Thus, SVC is a weakened version of AC; intuitively, it says that “the failure of AC to hold is parametrized by a mere set” (rather than a proper class).

At least assuming classical logic, it is equivalent to say that every set is a quotient of A×SA\times S, for we can extend any function defined on a subset of A×SA\times S to be constant on the complement of this subset.

Implications

The following statements are all consequences of SVC (some requiring excluded middle).

Models

SVC holds in all “ordinary” permutation? and symmetric model?s of (material) set theory, as well as in hereditarily-ordinal-definable and relatively-constructible submodels. However, it can fail in permutation models over proper classes.

Mike Shulman: The above facts make it sound plausible that SVC might hold internally in all Grothendieck toposes constructed from a base topos satisfying AC, but Makkai claims that it does not (although its consequence, the (local) axiom of small cardinality selection, does).

SVC and COSHEP

SVC can be regarded as a sort of “dual” or “complement” to COSHEP, since it deals with choice sets and COSHEP deals with projective sets. And while COSHEP implies the existence of projective resolutions, SVC implies the existence of (at least some) injective resolutions.

Moreover, at least assuming classical logic, SVC + COSHEP implies AC. First note that if SVC holds with SS, then it also holds with SS' for any surjection SSS'\to S; thus under SVC + COSHEP we may assume that SVC holds with a projective set SS. Let ZZ be any set, and let g:A×SZ Sg\colon A\times S \twoheadrightarrow Z^S be a surjection, where AA is choice. For each sSs\in S, let Z s={g(a,s)(s)aA}ZZ_s = \{g(a,s)(s) | a\in A\} \subseteq Z. Then each Z sZ_s is a surjective image of AA, and hence choice. If each ZZ sZ\setminus Z_s is nonempty, then (since SS is projective) we have a function f:SZf\colon S\to Z such that f(s)Z sf(s)\notin Z_s for all ss. But gg is surjective, so there exist s 0Ss_0\in S and a 0Aa_0\in A with g(a 0,s 0)=fg(a_0,s_0) = f, whence g(a 0,s 0)(s 0)=f(s 0)Z s 0g(a_0,s_0)(s_0) = f(s_0) \notin Z_{s_0}, a contradiction. Thus some Z sZ_s is all of ZZ, and hence ZZ is choice; so AC holds.

References

SVC was introduced in the paper

  • Andreas Blass, “Injectivity, projectivity, and the axiom of choice.” Trans. Amer. Math. Soc. 1979

where most of the above results were proven. Some others can be found in

  • Rathjen and Lubarsky, “Choice principles in constructive and classical set theories”.

Revised on September 5, 2011 16:16:53 by Toby Bartels (75.88.82.16)