Here we introduce Tarski-Grothendieck (TG) set theory, named after mathematicians Alfred Tarski and Alexander Grothendieck. TG theory extends ZFC with the TG Axiom ax-groth, which states that for every set there is an inaccessible cardinal such that is not in . The addition of this axiom to ZFC set theory provides a framework for category theory, thus for all practical purposes giving us a complete foundation for "all of mathematics."
We first introduce the concept of inaccessibles, including weakly and strongly inaccessible cardinals (df-wina and df-ina respectively ), Tarski classes (df-tsk), and Grothendieck universes (df-gru). We then introduce the Tarski's axiom ax-groth and prove various properties from that.