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- Is cardinality of non-measurable sets ##\aleph_0## or ##2^{\aleph_0}##?

The interval ##[0,1]## of real numbers has a non-zero measure. The set of all rational numbers in the interval ##[0,1]## has zero measure. But there are also sets that are somewhere in between, in the sense that their measure is neither zero nor non-zero. They are sets for which measure is not defined. Such sets, for instance, appear in the Banach-Tarski paradox. With a desire to get some better intuition of such sets, I ask about their cardinality. Are such sets continuous with cardinality ##2^{\aleph_0}##, or discrete with cardinality ##\aleph_0##? My intuition tells me that they should be continuous, but I want a confirmation.

Or is it perhaps undecidable in the ZFC axioms, in the same sense in which it is undecidable whether exist sets with cardinality bigger than ##\aleph_0## and smaller than ##2^{\aleph_0}##?

Or is it perhaps undecidable in the ZFC axioms, in the same sense in which it is undecidable whether exist sets with cardinality bigger than ##\aleph_0## and smaller than ##2^{\aleph_0}##?