Tarski-Grothendieck set theory

The Tarski-Grothendieck set theory (TG) is a system of axioms for the set- theoretical foundation of mathematics . It consists of the extension of the Zermelo-Fraenkel set theory with the axiom of choice , which represent the most widespread bases of the axiom that every set is an element of a Grothendieck universe , the so-called axiom of unattainable sets (in French axiom des univers , in English axiom of universes ). Like the Zermelo-Fraenkel set theory, it is based on first-order predicate logic . In addition to its importance as an object of investigation in set theory, it is widely used today as a basis in parts of mathematics, such as category theory and algebraic geometry . It is named after Alfred Tarski and Alexander Grothendieck .

history

In 1938 Tarski introduced the axiom of unreachable sets and investigated the relationship with strongly unreachable cardinal numbers . In the notes of the fourth part of the Séminaire de géométrie algébrique du Bois Marie from 1963–1964, which was influential on algebraic geometry and initiated by Grothendieck , the axiom des univers was presented on behalf of the Nicolas Bourbaki collective and its applicability to category theory and algebraic geometry. The Mizar Mathematical Library uses the Tarski-Grothendieck set theory as a system of axioms.

Axiom of unreachable sets

The axiom of the unreachable sets can be formulated in one of the following ZFC equivalent possibilities:

• A Grothendieck universe exists for every set , so . ${\displaystyle x}$ ${\displaystyle U}$ ${\displaystyle x\in U}$
• There are cardinal numbers of any size that cannot be reached (that is, for each ordinal number there is at least an equally large cardinal number that cannot be reached; the cardinal numbers that cannot be reached are confinally in the class of ordinals, axiom of inaccessibles)

Axiomatization without recourse to ZFC

With a direct definition of the Tarski-Grothendieck set theory without recourse to ZFC, it is possible to save some axioms, an axiomatization is possible as follows, using first-order predicate logic with equality:

• Axiom of extensionality : If two sets contain the same elements, they are equal.

${\displaystyle \forall X\forall Y(\forall x\ (x\in X\leftrightarrow x\in Y)\rightarrow X=Y)}$

• Foundation axiom : A set that contains an element contains a set that is disjoint to it.

${\displaystyle \forall X(\exists z\ z\in X\rightarrow \exists Y(Y\in X\wedge \nexists x(x\in X\wedge x\in Y)))}$

• Substitution axiom : For every two-digit predicate that is unambiguous to the right , and for every set, one can construct another set that contains the elements for which there is an element in the set for which the predicate is true with them. It is an axiom scheme , i.e. for every two-digit predicate the following axiom is contained in the Tarski-Grothendieck set theory: ${\displaystyle \varphi (x,y)}$ ${\displaystyle \varphi (x,y)}$

${\displaystyle \forall {\mathcal {A}}((\forall x,y,z\varphi (x,y)\wedge \varphi (x,z)\rightarrow y=z)\to \exists X\forall x(x\in X\leftrightarrow \exists y(y\in {\mathcal {A}}\wedge \varphi (x,y))))}$

• Axiom of the unreachable sets: for every set there exists another set which contains the first set, which is a transitive set , in which the power set of each of its elements is contained, and in which every non-equipotent subset is contained.

${\displaystyle \forall z\exists y\forall x\forall x'(z\in y\wedge ((x\in y\wedge x'\subseteq x)\to x'\in y)\wedge (x\in y\to {\mathcal {P}}(x)\in y)\wedge (x\subseteq y\to (x\in y\vee x\approx y)))}$

There is a short notation for the equality of and and a short notation for the fact that a set is an element of , which contains exactly the subsets of as elements. ${\displaystyle x\approx y}$ ${\displaystyle x}$ ${\displaystyle y}$ ${\displaystyle {\mathcal {P}}(x)\in y}$ ${\displaystyle P(x)}$ ${\displaystyle y}$ ${\displaystyle x}$

The axiom of substitution includes the axiom of discard . Pair set axiom, union axiom and power set axiom result from the fact that through the axiom of unreachable sets every set is an element of a level (see also Scott's system of axioms ). The axiom of choice results from the fact that every set is an element of a set that contains each of its non-equipotent subsets.

Web links

• Andrzej Trybulec: Tarski Grothendieck Set Theory. (pdf) January 1990, accessed on December 14, 2013 . 
• Tarski's axiom. January 3, 2006, accessed December 16, 2013 . 
• Tarski Grothendieck Set Theory (TG). Retrieved December 16, 2013 . 

Individual evidence

1. ↑ [1]
2. ↑ [2]
3. ^ Frank R. Drake: Set Theory . An Introduction to Large Cardinals. North Holland, Amsterdam 1974, ISBN 0-7204-2200-0 , pp. 68 .
4. ↑ Tarski, 1938, pp. 85-86.