Ordinal and cardinal exponentiation use similar notation, but they don't work the same way. Though it's common to see statements like "ω is the same as \aleph_0," it's misleading, because 2^ω = ω, and thus has lesser cardinality than 2^\aleph_0.

See also the "warning" here:

http://en.wikipedia.org/wiki/Ordinal_arithmetic#Exponentiati...