nLab small violations of choice





The basis of it all

 Set theory

set theory

Foundational axioms

foundational axioms

Removing axioms



The axiom of small violations of choice, or SVC, introduced by 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.


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


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

When we move to constructive mathematics, however, SVC quickly becomes unreasonable, because in the absence of LEM there are usually very few choice objects. In particular (see MO:a/339314) Simon Henry argues that any Grothendieck topos satisfying both SVC and ¬LEM\neg LEM (i.e. ¬P,P¬P\neg \forall P, P\vee \neg P, which holds in many toposes such as Sh([0,1])Sh([0,1])) is trivial.


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) \vert 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.


SVC was introduced in the paper

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

  • Michael Rathjen, Choice principles in constructive and classical set theories, in: Logic Colloquium ‘02, pp 299-326, Lect. Notes Log., 27, (2006) (pdf)

Last revised on October 27, 2022 at 08:50:49. See the history of this page for a list of all contributions to it.