A mathematical proof of S. Shelah’s theorem on the measure problem and related results
Recently, S. Shelah proved that an inaccessible cardinal is necessary to build a model of set theory in which every set of reals is Lebesgue measurable. We give a simpler and metamathematically free proof of Shelah's result. As a corollary, we get an elementary proof of the following result (without choice axiom): assume there exists an uncountable well ordered set of reals, then there exists a non-measurable set of reals. We also get results about Baire property,K σ-regularity and Ramsey property.
KeywordsMeasure Problem Infinite Subset Rapid Filter Inaccessible Cardinal Baire Property
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- 2.A.R.D. Mathias,A remark on rare filters, Colloquia Mathematica Societatis Janos Bolyai, 10 —Infinite and finite sets, to Paul Erdos on his 60th birthday (A. Hajnal, R. Rado and V. T. Sos, eds.), North Holland, 1975.Google Scholar
- 4.G. Mokobodzki,Ultrafiltres rapides sur N. Construction d'une densité relative de deux potentiels comparables, Seminaire Brelot-Choquet-Deny (théorie du potentiel), n0 12, Paris, 1967/68.Google Scholar
- 8.J. Stern,Regularity properties of definable sets of reals, circulated notes, 1981.Google Scholar