Solovay's model: ZF + DC with every set of reals measurable

If ZFC together with "there exists an inaccessible cardinal" is consistent, then so is the theory

ZF + DC + "every set of reals is Lebesgue measurable" + "every set of reals has the Baire property" + "every uncountable set of reals contains a perfect set",

where DC is the axiom of dependent choice.

Solovay (1970) builds the model by Levy-collapsing an inaccessible cardinal $\kappa$ to $\omega_1$ and then passing to an inner model of the extension. His own model is the class of sets hereditarily definable from a countable sequence of ordinals; the inner model $L(\mathbb{R})$ of the same extension is the other standard choice and satisfies the same conclusions. The two are not being claimed here to be the same class. Every set of reals in the model is definable from a real and an ordinal, and each such set is shown to be measurable by a homogeneity argument about the collapse.

What this rules out. Every classical pathology of the real line that is proved by well-ordering $\mathbb{R}$ is therefore not available from ZF + DC alone, relative to the stated large-cardinal hypothesis: a Vitali set, a Bernstein set, a Hamel basis for $\mathbb{R}$ over $\mathbb{Q}$ and with it a discontinuous additive solution of Cauchy's functional equation, and the Banach-Tarski decomposition.

• Not proved in this library. Neither the Levy collapse nor the measurability argument is developed here, and no measure theory is developed here either, so even the statement "Lebesgue measurable" is being borrowed.

• What would prove it. Three tracks at once: forcing (the Levy collapse and its homogeneity), large cardinals (the inaccessible, which Shelah 1984: the inaccessible is needed for measurability, not for the Baire property ‡ shows is genuinely required for the measurability clause), and Lebesgue measure theory. None of the three exists in this library.

• Why it matters here. It is the sharpest available answer to "how much choice does the pathology of $\mathbb{R}$ actually cost". DC is enough for essentially all of classical analysis of sequences and limits, and this model says that DC alone still produces none of the non-measurable objects. So whenever a later page produces such an object through Zorn's lemma ↗ or a well-ordering, the use of full The Axiom of Choice ↗ is not laziness: the ledger in The choice ledger: what costs the Axiom of Choice and what does not ↗ can record it as irreducible.

• Conditional discipline. The hypothesis here is stronger than mere consistency of ZF: it is the consistency of ZFC plus an inaccessible cardinal, which is strictly stronger and is not provable from Con(ZFC). That extra hypothesis is needed for the measurability clause and, by Shelah, not for the Baire-property clause.