Extended set theory (XST) is an axiomatic set theory introduced by Childs (1977). Its fundamental membership relation
is ternary: 
means that 
is an element of 
with scope 
. The scopes make it possible to represent structures such
as tuples, records, and functions
without first encoding positions or field names as parts of ordered
pairs.
Most XST axioms are scoped analogs of axioms of Zermelo-Fraenkel set theory, and XST permits urelements. Its Klass axiom also allows certain very large collections to be sets when their nested scopes are suitably bounded. This provides a way to manage the distinction between sets and proper classes that arises in category theory.
Blass and Childs (2014) interpreted XST in ZFC together with the assertion that there are arbitrarily large inaccessible cardinals, thereby establishing the consistency of XST relative to that theory.
