The use of aleph-0 for countable sets is not discouraged. I think what you mean is that using aleph-1 to mean "of the same cardinality as the reals" is discoura

The use of aleph-0 for countable sets is not discouraged. I think what you mean is that using aleph-1 to mean "of the same cardinality as the reals" is discouraged, because it's only appropriate in models where the continuum hypothesis holds. There's a not-so-commonly-used notation using the Hebrew letter beth instead of alpha, where beth_0 = aleph_0 and beth_{n+1} = 2^beth_n (note for experts: and you do the obvious thing at limit ordinals). So the cardinality of the real numbers is beth-1, whatever model you're in.