A colleague of mine suggested the following weakening of the axiom of choice:



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> If $\mathscr{F} := \{F\_\alpha\}$ is a *well-ordered* family of non-empty sets (i.e., there is a bijection between $\mathscr{F}$ and some von Neumann ordinal), then there is a choice function $f$ with domain $\mathscr F$ such that $f(F\_\alpha) \in F\_\alpha$ for all $\alpha$.
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The intuition behind this axiom is that the well ordering allows us to imagine picking the representatives "one at a time," and at any stage there is always a clear candidate for the next $F\_\alpha$ to consider.


**Question:** How does this axiom compare to other known weak forms of the axiom of choice?


Clearly it implies countable choice, but does it imply anything interesting that countable choice does not?



Logical strength of "choice functions exist for well-ordered families"?
This principle is known in set-theoretic literature as $\forall \kappa AC\_\kappa$, where $AC\_\kappa$ refers to the existence of choice functions for well-ordered families of size $\kappa$. It is discussed in detail in Chapter 8 of Jech's textbook "The axiom of choice". The principle $\forall\kappa AC\_\kappa$ implies slightly more than the countable axiom of choice, namely it implies $DC\_\omega$, the *axiom of dependent choices*, which states that any set relation without terminal nodes has an infinite branch. But $\forall \kappa AC\_\kappa$ does not imply the generalization of $DC\_\omega$ to $\omega\_1$, $DC\_{\omega\_1}$, which allows $\omega\_1$-many dependent choices. On the other hand, the principle $\forall \kappa DC\_\kappa$ does turn out to be equivalent to $AC$. More precisely, $DC\_\kappa$ states that for any set $S$ and any relation $R$ on $S^{<\kappa}\times S$, if for every $\vec x$ in $S^{<\kappa}$, there is $x\in S$ such that $\vec x R x$, then there is $f:\kappa\to S$ such that $f|\alpha R f(\alpha)$. 


