The axiom of small violations of choice, or SVC, introduced by Andreas Blass (1979), asserts that
Note that if , 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 , for we can extend any function defined on a subset of to be constant on the complement of this subset.
The following statements are all consequences of SVC (some requiring excluded middle).
Assuming SVC with , AC holds as soon as is choice.
The regular extension axiom (REA).
The axiom of multiple choice (AMC).
There are “enough highly filtered ordinals,” in the sense that for any set , there is a limit ordinal such that there is no cofinal function . (This is actually a consequence of REA.)
The (local) axiom of small cardinality selection.
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 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 , then it also holds with for any surjection ; thus under SVC + COSHEP we may assume that SVC holds with a projective set . Let be any set, and let be a surjection, where is choice. For each , let . Then each is a surjective image of , and hence choice. If each is nonempty, then (since is projective) we have a function such that for all . But is surjective, so there exist and with , whence , a contradiction. Thus some is all of , and hence 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