False: the all-Baire-property model needs an inaccessible

The claim under refutation is the usual relative-consistency assertion that an inaccessible-cardinal hypothesis is needed to obtain the all-Baire-property model. To refute that requirement it suffices to produce the model relative to ZFC alone. This reading is weaker than, and must not be replaced by, the formal assertion that the target theory's consistency disproves the consistency of ZFC plus an inaccessible.

By The exact equiconsistency of ZFC and the all-Baire-property model, the theory ZF+DC plus "every set of reals has the Baire property" is equiconsistent with ZFC alone: in particular, $\mathrm{Con}(\mathrm{Z}\mathrm{F}\mathrm{C})⟶\mathrm{Con}(\mathrm{Z}\mathrm{F}+\mathrm{D}\mathrm{C}+\text{all BP})\mathrm{.}$ The construction therefore needs no inaccessible-cardinal assumption, which refutes the requirement fixed in step 1.1. Equiconsistency with ZFC by itself does not prove that the target consistency fails to imply the consistency of a stronger theory, and no such claim is used here.

The comparison with measurability is a separate calibration: by Exact equiconsistency of universal measurability and an inaccessible, universal Lebesgue measurability is equiconsistent with ZFC plus an inaccessible cardinal. This fact neither supplies a separating model nor, by itself, proves a strict nonimplication between the two bare consistency statements; no such inference is made here.

The semantic separation is also witnessed: by Shelah's model separates universal Baire property from universal measurability there is, relative to $\mathrm{Con}(\mathrm{Z}\mathrm{F}\mathrm{C})$, a model of ZF+DC in which every set of reals has the Baire property and some set of reals is not Lebesgue measurable. This shows that the two regularity assertions themselves separate; it is not offered as a proof that one formal consistency statement fails to imply another.

Steps 1.2 and 1.3 refute the alleged need to assume an inaccessible in the relative-consistency construction and identify the established equiconsistency calibrations; step 1.4 supplies the semantic contrast. No lower bound for measurability transfers to the Baire property, and no unproved nonimplication between bare consistency statements is asserted.