the question is clear and is axiomatic set theory question.the axiom of choice is innocent and true. but consequences like well ordering theorem or Banach Taraski paradox are debatable.. as the prrof of well ordering theorem depends on the power set axiom i propose to restrict the power set axiom as the power set of a set exists provided the set has cardinality leasser than or equal to the power set of the power set of N. will it  well ordering theorem and Banch tarski paradox . 

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