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Is true that the set of the Borelians in $\mathbb{R}$ has the same cardinality of $\mathbb{R}$? I need of the Continuum Hypothesis for to prove this?

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marked as duplicate by Andrés E. Caicedo, user61527, user1337, Thomas Andrews, Git Gud Aug 24 '13 at 23:24

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  • $\begingroup$ What do you mean by Borelians? Do you mean Borel sets? $\endgroup$ – user61527 Aug 24 '13 at 22:54
  • $\begingroup$ Yes, this is true, and no, the continuum hypothesis is not needed. It is also true that every Borel set either is countable or has the size of the reals (in fact, contains a perfect subset), and no, the continuum hypothesis is not needed to show this either, but the axiom of choice is required. $\endgroup$ – Andrés E. Caicedo Aug 24 '13 at 23:00

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