I'm in search of a set theory that satisfies the following requirements.
- There is a universal set $V$ such that $\forall x(x \in V)$. So for example, $V \in V$.
- Sets whose elements are 'large' exist. e.g. I want $\lbrace V,\emptyset\rbrace$ to be a well-defined set with cardinality $2$.
- [Edit] Sets form a boolean algebra; in particular, the complement of a set always exists.