Decomposing the real line into Borel sets closed under addition
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Publication:3460526
DOI10.1002/MALQ.201400100zbMATH Open1357.03079arXiv1406.0701OpenAlexW2129151842MaRDI QIDQ3460526
Publication date: 7 January 2016
Published in: Mathematical Logic Quarterly (Search for Journal in Brave)
Abstract: 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 and even in the theory if the number of pieces can be uncountable but less than the continuum. We also investigate various versions: what happens if we drop the Borelness requirement, if we replace addition by multiplication, if the pieces are subgroups, if we partition , and so on.
Full work available at URL: https://arxiv.org/abs/1406.0701
Descriptive set theory (03E15) Classes of sets (Borel fields, (sigma)-rings, etc.), measurable sets, Suslin sets, analytic sets (28A05) Consistency and independence results (03E35) Cardinal characteristics of the continuum (03E17)
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Related Items (2)
Invariant sets in real number fields ⋮ Decompositions of the positive real numbers into disjoint sets closed under addition and multiplication
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