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We consider decompositions of the real line into pairwise disjoint Borel pieces so that each piece is closed under addition. How many pieces can there be? We prove among others that the number of pieces is either at most 3 or uncountable, and we show that it is undecidable in ZF C and even in the theory ZF C + c = ω 2 if the number of pieces can be… (More)

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