Tarski–Grothendieck set theory

Tarski–Grothendieck set theory (TG, named after mathematicians Alfred Tarski and Alexander Grothendieck) is an axiomatic set theory. It is a non-conservative extension of Zermelo–Fraenkel set theory (ZFC) and is distinguished from other axiomatic set theories by the inclusion of Tarski's axiom, which states that for each set there is a Grothendieck universe it belongs to (see below). Tarski's axiom implies the existence of inaccessible cardinals, providing a richer ontology than ZFC. For example, adding this axiom supports category theory.

The Mizar system and Metamath use Tarski–Grothendieck set theory for formal verification of proofs.