In 1970, Solovay created a model in which all sets of reals are Lebesgue measurable. His argument involved an inaccessible cardinal, leaving it an open question whether such a model could be built without using such a strong axiom. In 1984, Shelah showed that the inaccessible cardinal is necessary. We will be proving Shelah's result, and the majority of the talk will focus on an ingenious construction of a nonmeasurable set.
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