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There are types of sets. a) A set such that no subset of it belongs to it. A={x|x is not a subset of } Cantor theorem appears to be applicable on it. b) A set such that each of its element is a subset of it self. B={y|y is a subset of B} Obviously such a set is only possible if either there are no URELEMENTS or at least there are no URELEMENTS in the set. Let y1 belongs to the set B THEN y1 is a subset of B by def.Let z1 belongs to y1 then z1 belongs to B ,SINCE IT IS AN ELEMENT OF SUBSETy1. But again a1 belongs to z1 and so on. Cantor theorem is not applicable on it. c)A SET C such that either its element is a subset or not a subset. C={s,t| s is not a subset of of C,t isa subset of C} Cantor theorem is not applicable on it,since P(C) = {p| p is a subset of C} a BIJECTION IS NOT POSSIBLE/ A BIJECTION DOES NOT EXIST FROM C to P(C) d) Set of All sets. Each and everey subset of set of all set belongs to it. So Cantor"s theorem ios not valid. When Cantor proved rather attemted to prove his theorem there were no MKS,ZFC etc theories

NONCANTORIANSETS  

SETS WHERE CANTOR THEOREM IS NOT VALID

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