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 ?
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Dependent choice, for example. Or choice for wellordered families, see Existence of model of ZF without AC, but with many choice function 


There are many many many axioms. 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 nonempty 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 $\mathsf{DC}_\kappa$. There are many choice principles and they come in different flavours and shapes. Some good places to start would be:


