This question may appear banal, but there seems to be more than meets the eye; a common glitch is to explain numbers by the "size" of sets without saying how to measure or compare the size of sets.

Can numbers be defined via Set theory alone, i.e. using only operations on sets or is it inevitable to use additional operations (careful here with equality of sets, because the cardinality could be the same, but the kind of elements may be different)?