Past exam of the mathematics course of the University of Cambridge / 2001 / ia / Paper 4 / 1E (Numbers and Sets)

Suppose $F:X\to \mathcal{P}(X)$ were a surjection onto the power set of the set $X$. Form the subsetSurjectivity would give $a\in X$ with $F(a)=D$. But then $a\in D$ holds exactly when $a\notin F(a)=D$, a contradiction. Hence no such surjection exists, and in particular there is no bijection between a set and its power set. This is the diagonal construction in Cantor's theorem; it also covers the empty set, whose power set has one element.

No such set exists. If $R$ had precisely the sets $x$ satisfying $x\notin x$ as its elements, applying that membership criterion to $R$ itself would givea contradiction. This is Russell's paradox. It rules out this unrestricted universal collection; a set whose elements happen not to contain themselves is perfectly possible, for example the empty set.