It seems that in most theorems outside of set theory where the size of some set is used in the proof, there are three possibilities: either the set is finite, countably infinite, or uncountably infinite. Are there any well known results within say, algebra or analysis that require some given set to be of cardinality strictly greater than $2^{\aleph_{0}}$? Perhaps in a similar vein, are any objects encountered that must have size larger than $2^{\aleph_{0}}$ in order for certain properties to hold?