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For context, Zermelo Frankel Set Theory with choice is typically what are taken to be the basic axioms of mathematics. But one alternative model is the Solovay Model. In this model, the axiom of choice is limited to countable choice and the model includes an inaccessible cardinal. It was shown that, in this model, all subsets of the reals are measurable. My question then is what else is different in this model and why isn’t it more popular. There are straight forward things like there would now be vector spaces without a basis, but I’m curious what other implications this would hav
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