Cantor’s Paradox demonstrates that there is no “set of all sets”. We know that every
set has a cardinality, and so it’s natural to ask if there is a set of “set of all possible
cardinalities”.
The answer to this is also no. Your task is to prove it.
Let X be a set whose elements are sets, and suppose X has the property that every
set is the same cardinality as some element of X. In other words, suppose X is a set
that contains a “representative” of every possible cardinality.
Show that X can’t exist.