638 名前：rom Wikipedia, the free encyclopedia In the mathematical field of set theory, the Solovay model is a model constructed by Robert M. Solovay (1970) in which all of the axioms of Zermelo?Fraenkel set theory (ZF) hold, exclusive of the axiom of choice, but in which all sets of real numbers are Lebesgue measurable. The construction relies on the existence of an inaccessible cardinal.
In this way Solovay showed that the axiom of choice is essential to the proof of the existence of a non-measurable set, at least granted that the existence of an inaccessible cardinal is consistent with ZFC, the axioms of Zermelo?Fraenkel set theory including the axiom of choice.
Construction
Instead of using Solovay's model N, one can also use the smaller inner model L(R) of M[G], consisting of the constructible closure of the real numbers, which has similar properties.
Complements
See Raisonnier (1984) and Stern (1985) and Miller (1989) for expositions of Shelah's result.
Shelah & Woodin (1990) showed that if supercompact cardinals exist then every set of reals in L(R), the constructible sets generated by the reals, is Lebesgue measurable and has the Baire property; this includes every "reasonably definable" set of reals. (引用終り)
Solovay 1970 原論文 https://www.math.wisc.edu/~miller/old/m873-03/solovay.pdf A Model of Set-Theory in Which Every Set of Reals is Lebesgue Measurable Robert M. Solovay The Annals of Mathematics, 2nd Ser., Vol. 92, No.1 (Jul., 1970), 1-56.
http://www.math.wisc.edu/~miller/index.html Arnold W. Miller []
