# A model of Krivine

In a paper by J.-L. Krivine, Modèles de ZF+AC dans lesquels tout ensemble de réels définissable en termes d'ordinaux est mesurable-Lebesgue [C. R. Acad. Sci. Paris Sér. A-B 269 (1969), A549–A552, MR0253894], he presents a model of ZFC in which every set of reals definable from ordinals is Lebesgue measurable.

My question is:

1. He seems not to assume the existence of an inaccessible cardinal in the ground model, is this correct?

2. In Theorem 3 of that paper, he concludes that every set of reals definable from ordinals is Lebesgue measurable. Does this continue to hold if we replace definable from ordinals by definable from a countable sequence of ordinals?

3. Does Theorem 3 of that paper still hold if we replace "Lebesgue measurable" by "having the property of Baire"?

-

1. Yes, that is correct. Krivine's observation is that by collapsing the continuum (or any larger cardinal) to $\aleph_0$ using finite conditions then any set of reals definable from ground model parameters (e.g. ordinals) is Lebesgue measurable in the extension.