Does Hahn-Banach yield a Hamel basis for $\mathbb{R}$ over $\mathbb{Q}$? (open)

Work in ZF, without the axiom of choice. Write HB for the Hahn-Banach theorem: if $p$ is a sublinear functional on a real vector space $X$ and $f$ is a linear functional on a subspace of $X$ dominated by $p$, then $f$ extends to a linear functional on all of $X$ still dominated by $p$.

Question. Does HB imply that $\mathbb{R}$, as a vector space over $\mathbb{Q}$, has a basis?

Status: open. It is recorded as an open question in Howard and Rubin's catalogue of consequences of the axiom of choice, as the implication from their form for Hahn-Banach to their form for a Hamel basis of $\mathbb{R}$ over $\mathbb{Q}$. No proof of the implication and no model of ZF separating the two is known.

Not proved in this library, and not proved anywhere. This library does not develop functional analysis and does not build models of ZF, so neither half of the question is reachable here; both belong to deferred tracks. Nothing on any page depends on the answer.

What is known, and what would settle it. The endpoints are well understood. Full choice gives a Hamel basis, since "every vector space has a basis" is equivalent to the axiom of choice (The Axiom of Choice ↗), the standard proof running through Zorn's lemma (Zorn's lemma ↗). In the other direction, granted the consistency of ZF, HB is not a theorem of ZF + DC. The cheapest route to that, and the one this item relies on, does not go through Lebesgue measure. HB applied to a nonzero element of $\ell^\infty / c_0$ produces a nonzero linear functional on that space, and from such a functional one gets a set of reals without the Baire property. Shelah (1984) showed that Solovay's inaccessible can be dispensed with for the Baire property, so the consistency of ZF alone yields a model of ZF + DC in which every set of reals has the Baire property; in that model $(\ell^\infty / c_0)^*$ is trivial, so HB fails there. This is worth spelling out because the obvious argument is more expensive: HB also implies, in ZF, the existence of a non-Lebesgue-measurable set (Foreman and Wehrung, 1991) and indeed the Banach-Tarski paradox (Pawlikowski, 1991), but a model of ZF + DC in which every set of reals is measurable costs an inaccessible cardinal, so that route would only give the unprovability of HB relative to a large cardinal.

HB is also, granted Con(ZF), strictly weaker than choice: the Boolean prime ideal theorem implies it outright (Luxemburg, 1969), and BPI does not imply the axiom of choice (Halpern and Lévy, 1971), so neither does HB. Pincus (1974) proved the sharper separation that HB does not imply BPI, refuting the prevailing conjecture of the 1960s. All of these are relative-consistency results and nothing stronger. A Hamel basis for $\mathbb{R}$ over $\mathbb{Q}$ likewise yields a non-measurable set. So the two statements sit strictly between ZF and AC, on those cited results and under the consistency of ZF, and the question is how they are ordered with respect to each other. Settling it means either deriving a Hamel basis from HB in ZF, or producing a model of ZF in which HB holds and $\mathbb{R}$ has no basis over $\mathbb{Q}$.

The nearest recent progress is a separation of the two classical consequences of a Hamel basis from each other: Larson and Shelah (2026) construct a model of ZF + DC containing a discontinuous additive endomorphism of $\mathbb{R}$ but no Hamel basis for $\mathbb{R}$. That does not touch HB, but it shows the two targets in this and the companion question are genuinely different targets and not notational variants.

A note on the reference. The form numbers used for these questions in the working notes are $52$ for Hahn-Banach, $367$ for a Hamel basis of $\mathbb{R}$ over $\mathbb{Q}$ and $366$ for a discontinuous additive function, taken from the Howard-Rubin numbering. These three numbers are unverified. The Consequences of the Axiom of Choice project database that hosted the searchable numbering, and the later mirror of it, both fail to answer as of 2026-07-26, and no other online source consulted lists the numbering, so they could not be re-checked; the book remains the reference and the numbers should be treated as a pointer into it rather than as a verified citation.

Why it matters here. This library keeps an explicit ledger of what each result costs in choice, and the ledger is supposed to be exact. This entry is a place where exactness is impossible: the cost of one of the most-used theorems in analysis, measured against one of the most-used pathologies in analysis, is not known. That is worth stating rather than rounding off to "both need choice".