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It is well known that even if one works with say ZFC as a base theory, Grothendieck topoi do not in general satisfy even fairly weak axioms like countable choice or small violations of choice and one ...

I just read Dr Strangechoice's explanation that if all subsets of the real numbers are Lebesgue measurable, then you can partition $2^\omega$ into more than $2^\omega$ many pairwise disjoint nonempty ...