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Abstract

We let: ZF = the Zermelo-Fraenkel axioms of set theory without the Axiom of Choice„(AC) . ZFC = ZF + AC . I = " There exists an inaccessible cardinal " . ψ = " Every set of reals definable from a count able sequence of ordinals is Lebesgue measurable ". DC = the Axiom of Dependent Choices. LM = " Every set of reals is Lebesgue measurable In 1970, Solovay published a proof by forcing of the following relative consistency result: Theorem If there exists a model M of ZFC + I, then there exist extensions M [G] and N of M such that: (a) M [G] |= ZFC + ψ. (b) N I= ZF + DC + LM . Boolean-valued techniques are used here to retrace Solovay's proof on a different foundation and prove the following result: Theorem Let IK be a non-minimal standard transitive model of ZFC + I. Then: (a) IK |= there is a model of ZFC + ψ (b) IK |= there is a model of ZF + DC + LM .