# Reverse-mathematical strength of Banach-Tarski

What is the reverse mathematical strength of the Banach-Tarski paradox?

The usual proof of Banach-Tarski should carry out in $$\mathrm{ZF}+\mathrm{AC}_\kappa$$, where $$\kappa$$ is the supremum of the sizes of partitions of $$\mathbb R$$, and $$\mathrm{AC}_\kappa$$ is choice for families of sets of size $$\kappa$$. However I am mainly focusing on replacing $$\textrm{ZF}$$ here with a weaker theory.

All machinery appearing in the proof seems to have low rank in the von Neumann hierarchy, so the axiom of powerset does not seem to have to be iterated many times. (In contrast it is a result of Friedman that Borel determinacy requires $$\omega_1$$-fold iteration of powerset applies to $$\mathbb N$$ in order to prove, although its statement only refers to sets of low rank. Yet all machinery used in proving Banach-Tarski seems to have low rank.) For example, if $$\mathcal P(\mathbb N)$$ is identified with the reals, the unit ball will be a subset of $$\mathcal P(\mathcal P(N))$$, a finite set of pieces of the unit ball will be codable as a member of $$\mathcal P(\mathcal P(\mathcal N))$$ using a coding function $$[\mathcal P(\mathcal P(\mathcal N))]^{<\aleph_0}\to \mathcal P(\mathcal P(\mathcal N))$$ (where $$[\mathcal P(\mathcal P(\mathcal N))]^{<\aleph_0}$$ is the set of finite subsets of $$\mathcal P(\mathcal P(\mathcal N))$$), and a quotient of the unit ball will be identified with $$\mathcal P(\mathcal P(\mathcal P(\mathbb N)))$$.

Some results are known about theories being sufficient or not sufficient to prove Banach-Tarski, although most are about choice rather than replacing the $$\mathrm{ZF}$$ part of the theory:

• It is known that the Hahn-Banach theorem implies the Banach-Tarski paradox. (Pawlikowski, "The Hahn-Banach theorem implies the Banach-Tarski Paradox", 1989), proven with base theory $$\mathrm{ZF}$$.
• It is known that Banach-Tarski is implied by the statement "every partial preorder extends to a linear preorder that preserves strict order" (MO question #150945), but this is proven with base theory $$\mathrm{ZF}$$.
• Banach-Tarski is number HR 309 in Howard and Rubin's book Consequences of the Axiom of Choice. There is an MO question "Equivalence of the Banach–Tarski paradox" relevant to this result.
• Banach-Tarski is known to not be provable in $$\mathrm{ZF}+\mathrm{DC}$$ if it is consistent, where $$\mathrm{DC}$$ is the axiom of dependent choice. (Corollary 13.3 of Wagon, The Banach-Tarski Paradox, 1993)
• It's not too clear what you are asking. Are you asking whether Banach-Tarski reverses to some other standard axiom/theorem over some system weaker than ZF? Or are you asking what the minimal version of Choice is that would be needed to prove Banach-Tarski? Dec 13, 2023 at 13:09
• @MikhailKatz When originally posting I may have had too broad of a question in mind, with the intended goal being to optimize on "two fronts" (both weakening the base theory and weakening the choice assumption), rather than a reversal to a standard theorem. This may be too ambitious, so I will refocus and rephrase the post. The question of weakening the necessary choice assumption seems to already be better-documented, so I will ask about weakening the base theory.
– C7X
Dec 13, 2023 at 17:55
• C7X, thanks for the clarification. You may want to elaborate a bit on the interesting issue concerning ZF+DC. Here as you know Solovay is not sufficient because of the IC hypothesis, but an additive total measure can be constructed anyway. Dec 14, 2023 at 7:38