Tarski–Grothendieck set theory

The Tarski - Grothendieck set theory (TG ) is an axiom system for set-theoretic foundations of mathematics. It consists of the extension of the Zermelo -Fraenkel set theory with the axiom of choice, which are the most common basics to the axiom that any amount is an element of a Grothendieck universe, the so-called axiom of inaccessible quantities ( in French axiom of the univers, in English axiom of universes ). Just as the Zermelo -Fraenkel set theory it is based on first order predicate logic. Apart from its importance as a subject of set theory it is used as a basis today in parts of mathematics, such as category theory and algebraic geometry, wide use. It is named after Alfred Tarski and Alexander Grothendieck.

History

In 1938 Tarski an axiom of the inaccessible volumes and examined the relationship with strongly inaccessible cardinal numbers. In the notes of the fourth part of the influential on algebraic geometry, initiated by Grothendieck Séminaire de géométrie algébrique du Bois Marie of 1963-1964, the axiom of the univers were presented on behalf of the collective Nicolas Bourbaki and its applicability to category theory and algebraic geometry. The Mizar Mathematical Library uses the Tarski - Grothendieck set theory as an axiom system.

Axiom of inaccessible quantities

The axiom of inaccessible quantities can be formulated in one of the following equivalent in the context of ZFC options:

• For each set, there exists a Grothendieck universe so.
• There exist arbitrarily large strongly inaccessible cardinal numbers (ie, for each ordinal exists at least as large strongly inaccessible cardinal number that strongly inaccessible cardinals are cofinal in the class of ordinal numbers, axiom of inaccessibles )

Axiomatization without resorting to ZFC

In a direct definition of the Tarski - Grothendieck set theory without recourse to ZFC, it is possible to save some axioms, a axiomatization is approximately as follows possible, with first-order logic with equality will be used:

• Extensionality: if two sets contain the same elements, they are the same.

• Foundation axiom: A set containing an element that contains a disjoint to her lot.

• Substitution axiom: For each binary predicate, which is quite unique, and for any amount, one can construct another set containing the elements for which the set is an element for which the predicate is true with them. It is an axiom schema, that is, for each binary predicate is the following axiom included in the Tarski - Grothendieck set theory:

• Axiom of inaccessible quantities: For each set there is a different amount, in the first set is included, which is a transitive set, in which the power set of each of its elements is included, and is included in which each non- equally powerful subset.

The substitution axiom includes the axiom of separation. Pair axiom, Union axiom and power set axiom arising from the fact that by the axiom of inaccessible quantities lots of element of a stage ( see also Scott cal system of axioms ). The axiom of choice follows from the fact that any amount is member of a set that contains each of its non-uniformly powerful subsets.