I am working on the Banach-Tarski paradox and the fact that the Hahn-Banach theorem implies that paradox. The proof involves the equivalence of the Hahn-Banach theorem and the fact that for every Boolean algebra $\mathcal A$ there is a finitely additive "measure" $\mathrm m:\mathcal A \to [0,+\infty]$ such that some element of $\mathcal A$ has finite measure). $\mathsf{ZF}$ is not enough to derive the Banach-Tarski paradox, nor is $\mathsf{ZF}+\mathsf{DC}$.

However, I am interested in reversing the roles, in order to characterize the Banach-Tarski paradox in terms of choice-like axioms. Hahn-Banach is strictly weaker than the axiom of choice or the ultrafilter lemma/boolean prime ideal theorem.

Can we find a weaker axiom to derive the Banach-Tarski's paradox? Do you know any reference or have any ideas on this? Are there weaker axioms that should suffice in order to get the Banach-Tarski paradox? Is it possible to reformulate the Banach-Tarski paradox in a way that this assumption allows us in $\mathsf{ZF}$ to recover the Hahn-Banach extension theorem?

These questions do not seem clear to me, so I would be very pleased if they are somehow absurd. Thanks in advance.

Consequences of the Axiom of Choice? That's where people always send me whenever I ask a question like this. $\endgroup$ – Nate Eldredge Mar 24 '16 at 1:42