>>284 補足

Vitali_setを否定するSolovay modelでは、DC（dependent choice）が使われる
”The axiom of dependent choice implies the axiom of countable choice and is strictly stronger.”

(参考)
https://en.wikipedia.org/wiki/Solovay_model
Solovay model
Statement
ZF stands for Zermelo–Fraenkel set theory, and DC for the axiom of dependent choice.
Solovay's theorem is as follows. Assuming the existence of an inaccessible cardinal, there is an inner model of ZF + DC of a suitable forcing extension V[G] such that every set of reals is Lebesgue measurable, has the perfect set property, and has the Baire property.

https://en.wikipedia.org/wiki/Axiom_of_dependent_choice
Axiom of dependent choice
Relation with other axioms
The axiom of dependent choice implies the axiom of countable choice and is strictly stronger.[4][5]

https://en.wikipedia.org/wiki/Vitali_set
Vitali_set
Properties
No Vitali set has the property of Baire.[2]
By modifying the above proof, one shows that each Vitali set has Banach measure
0. This does not create any contradictions since Banach measures are not countably additive, but only finitely additive