# The small cardinality selection axiom

## Idea

The small cardinality selection axiom (SCSA) is a weak form of the axiom of choice, which asserts in a certain precise way that choice fails “in at most a small way”. It was introduced by Michael Makkai for the study of anafunctors, and thus it also has consequences for the existence of stack completions.

## Definition

The “global version” of the axiom states:

Therefore, we have a class function assigning to each set $A$, another set ${\Vert A\Vert}$ and a bijection $A \cong {\Vert A \Vert}$, in such a way that ${\Vert{-}\Vert}$ takes only set-many values on each isomorphism class in Set.

## Relationship to other forms of choice

• In the presence of the axiom of global choice in material set theory, the category Set has a skeleton, namely the category of von Neumann ordinals. Ordinary AC is all that is needed to ensure that every set is bijective to a von Neumann ordinal, and that ordinal can be uniquely determined as the smallest in its cardinality class, but in order to additionally choose for every set a bijection to its cardinality, we need global AC.

• SCSA also follows from the “global” version of the axiom of small violations of choice, as proven in Makkai’s paper (attributed to the referee).

Mike Shulman: Is there a “local” version of SCSA which follows from non-global AC and non-global SVC? Is there a structural version?

## References

Revised on July 19, 2011 00:46:42 by Toby Bartels (76.85.192.183)