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Given infinite sets $X$ and $Y$ such that $X$ has strictly smaller cardinality than $Y$, is it the case that the power set of $X$ has strictly lower cardinality than the power set of $Y$?

Andrés E. Caicedo
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Nishant
  • 9,155
  • I believe this is undecidable from ZFC, but someone (Asaf) with more knowledge should provide the answer. – Thomas Andrews Jul 15 '14 at 00:45
  • More discussion at: http://mathoverflow.net/questions/17152/when-2a-2b-implies-a-b-a-b-cardinals – Shawn O'Hare Jul 15 '14 at 00:56
  • Consider: http://math.stackexchange.com/questions/143452/do-sets-whose-power-sets-have-the-same-cardinality-have-the-same-cardinality?lq=1 and http://math.stackexchange.com/questions/29366/do-sets-whose-power-sets-have-the-same-cardinality-have-the-same-cardinality which are about the contrapositive of this statement – Vincent Pfenninger Jul 15 '14 at 00:57

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