User blog comment:Rgetar/Help to name function/@comment-30754445-20180423200016/@comment-30754445-20180428201112

More precisely, 2ℵ₀ denotes the cardinality of the set of all functions from ℵ₀ to 2.

Since ℵ₀ = ω = {0,1,2,3,...} and 2= {0,1} then 2ℵ₀ denotes the set of functions from {0,1,2,3,...} to {0,1}.

That's how cardinal exponents are defined. And it is, indeed, easy to show that the cardinality of the powerset of A (for any set A) is also 2card(A) by this definition: The advantage of the definition in the previous paragraph, however, is that it works for bases bigger than 2.