  There is a widespread misapprehension that Cantor’s theorem shows that there cannot be a consistent set theory with a universal set. That error can be refuted by Jensen’s construction of his models for NFU, but by far the most appealing demonstration is a later construction of a model of a set theory with a universal set by Church and Oswald independently and roughly simultaneously in the early 1970s. It has the advantage over NFU of respecting extensionality: it is a genuine model of a theory of pure sets. The construction is a refinement of Rieger-Bernays methods. It goes as follows: start with a model of something like ZF, and a bijection \(k\) between \(V\) and \(V \times \{0, 1\}\). We now define a new membership relation \(E\) so that \(xEy\) iff either the second component of \(k(y)\) is 1 and \(x \in\) first component, or the second component of \(k(y)\) is 0 and \(x \not\in\) first component. \(k^{-1}(\varnothing)\) is now the universal set. In fact  \(k^{-1}(\langle x,0\rangle)\) and  \(k^{-1}(\langle x,1\rangle)\) are set-complements in this new structure.