Shelah 1984: the inaccessible is needed for measurability, not for the Baire property

Shelah (1984) settles which half of Solovay's model: ZF + DC with every set of reals measurable ‡ really needs a large cardinal.

If ZF is consistent, then so is ZF + DC + "every set of reals has the Baire property". No inaccessible cardinal is required for this half.

The measurability half is different. If ZF + DC + "every set of reals is Lebesgue measurable" is consistent, then so is ZFC + "there exists an inaccessible cardinal". So the large-cardinal hypothesis in Solovay's theorem is not an artefact of the proof: the two theories are equiconsistent, and the inaccessible cannot be removed.

• Not proved in this library. Neither direction is proved here. The second direction in particular is a reverse-mathematical calculation about consistency strength, of a kind this library has no machinery for.

• What would prove it. For the first direction, an amalgamated Souslin forcing replacing the Levy collapse. For the second, the observation that if every set of reals is measurable then $\omega_1$ is inaccessible in $L[r]$ for every real $r$, which needs the fine structure of the constructible universe relativised to a real. Both belong to a forcing and inner-model track that this library does not contain.

• Why it matters here. It sharpens the reading of Solovay's model: ZF + DC with every set of reals measurable ‡ in a way that matters for how the library reports costs. "Every set of reals has the Baire property" is consistent with ZF + DC at no extra consistency cost at all, so any Baire-category pathology on $\mathbb{R}$ that this library later produces must be using more than DC. Measurability pathologies sit strictly higher. When The choice ledger: what costs the Axiom of Choice and what does not ↗ records what a theorem costs, this is the result that keeps the two cases apart.

• Conditional discipline. The first clause is relative to Con(ZF); the second is an implication between consistency statements and is not an assertion that an inaccessible cardinal exists.