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 that of conventional set theories such as ZFC. For example, adding this axiom supports category theory.
Tarski-Grothendieck set theory starts with conventional Zermelo-Fraenkel set theory and then adds "Tarski's axiom". We will use the axioms, definitions, and notation of Mizar to describe it. Mizar's basic objects and processes are fully formal; they are described informally below. First, let us assume that:
TG includes the following axioms, which are conventional because they are also part of ZFC:
It is Tarski's axiom that distinguishes TG from other axiomatic set theories. Tarski's axiom also implies the axioms of infinity, choice, and power set. It also implies the existence of inaccessible cardinals, thanks to which the ontology of TG is much richer than that of conventional set theories such as ZFC.
- every subset of every member of ;
- the power set of every member of ;
- every subset of of cardinality less than that of .
That looks much like a "universal set" for - it not only has as members the powerset of , and all subsets of , it also has the powerset of that powerset and so on - its members are closed under the operations of taking powerset or taking a subset. It's like a "universal set" except that of course it is not a member of itself and is not a set of all sets. That's the guaranteed Grothendieck universe it belongs to. And then any such is itself a member of an even larger "almost universal set" and so on. It's one of the strong cardinality axioms guaranteeing vastly more sets than one normally assumes to exist.
The Mizar language, underlying the implementation of TG and providing its logical syntax, is typed and the types are assumed to be non-empty. Hence, the theory is implicitly taken to be non-empty. The existence axioms, e.g. the existence of the unordered pair, is also implemented indirectly by the definition of term constructors.
The system includes equality, the membership predicate and the following standard definitions:
The Metamath system supports arbitrary higher-order logics, but it is typically used with the "set.mm" definitions of axioms. The ax-groth axiom adds Tarski's axiom, which in Metamath is defined as follows:
? ?y(x ? y ? ?z ? y (?w(w ? z -> w ? y) ? ?w ? y ?v(v ? z -> v ? w)) ? ?z(z ? y -> (z ? y ? z ? y)))