# what axioms are between AC and Countable choice !

Hi All. Need some information. We all know axiom of choice (AC) and countable choice. Which axioms are between these two. I mean weaker that Axiom of choice but stronger than countable choice ?

-
What do you need this information for? Do you have a specific theorem that follows from ZFC but not from ZF+Countable choice? –  Goldstern Jan 4 '13 at 20:46

Dependent choice, for example. Or choice for well-ordered families, see Existence of model of ZF without AC, but with many choice function

-
For example you have the Principle of Dependent Choice, you also have the generalized axiom of choice for $\kappa$, "For every family of size $\kappa$ of non-empty sets there is a choice function." and whenever $\kappa>\aleph_0$ this is a strictly stronger axiom. In a similar fashion the principle of dependent choice is extended to what is known as $\mathsf{DC}_\kappa$.