Question

I'm in search of a set theory that satisfies the following requirements.

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

Answer 1

Here is a CW answer incorporating the answers in the comments. Both NF (New Foundations) and NFU (New Foundations with urelements, obtained from NF by weakening the axiom of extensionality) satisfy conditions (1) through (3).