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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.

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  • Tarski-Grothendieck-Mengenlehre
  • Tarski–Grothendieck set theory
  • Teoria degli insiemi di Tarski-Grothendieck
  • Teoria dos conjuntos de Tarski-Grothendieck
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  • Die Tarski-Grothendieck-Mengenlehre (TG) ist ein Axiomensystem zur mengentheoretischen Grundlegung der Mathematik. Sie besteht aus der Erweiterung der Zermelo-Fraenkel-Mengenlehre mit Auswahlaxiom, welche die verbreitetsten Grundlagen darstellen, um das Axiom, dass jede Menge Element eines Grothendieck-Universums ist, das sogenannte Axiom der unerreichbaren Mengen (im Französischen axiom des univers, im Englischen axiom of universes). Ebenso wie die Zermelo-Fraenkel-Mengenlehre basiert sie auf der Prädikatenlogik erster Stufe. Neben ihrer Bedeutung als Untersuchungsgegenstand der Mengenlehre findet sie als Grundlagen heute in Teilen der Mathematik, etwa der Kategorientheorie und der algebraischen Geometrie, weite Verwendung. Sie ist nach Alfred Tarski und Alexander Grothendieck benannt.
  • La teoria degli insiemi di Tarski-Grothendieck (TG) è una teoria assiomatica degli insiemi così chiamata in riferimento ai matematici Alfred Tarski e Alexander Grothendieck. Essa è caratterizzata dall'Assioma di Tarski ed è un'estensione non-conservativa della teoria degli insiemi di Zermelo - Fraenkel.
  • 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.
  • A teoria dos conjuntos de Tarski-Grothendieck (TG, assim denominada em referência aos matemáticos Alfred Tarski e Alexander Grothendieck) é uma teoria dos conjuntos axiomática. Também é uma extensão não conservativa da teoria dos conjuntos de Zermelo-Fraenkel (ZFC) e pode ser distinguida de outras teorias dos conjuntos axiomáticas por causa da inclusão do axioma de Tarski, que diz que para cada conjunto existe um universo de Grothendieck a qual pertence. O axioma de Tarski implica na existência de cardinais inacessíveis, fornecendo uma ontologia mais rica do que outras teorias como a ZFC. Por exemplo, adicionar esse axioma dá suporte a teoria das categorias.
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  • Die Tarski-Grothendieck-Mengenlehre (TG) ist ein Axiomensystem zur mengentheoretischen Grundlegung der Mathematik. Sie besteht aus der Erweiterung der Zermelo-Fraenkel-Mengenlehre mit Auswahlaxiom, welche die verbreitetsten Grundlagen darstellen, um das Axiom, dass jede Menge Element eines Grothendieck-Universums ist, das sogenannte Axiom der unerreichbaren Mengen (im Französischen axiom des univers, im Englischen axiom of universes). Ebenso wie die Zermelo-Fraenkel-Mengenlehre basiert sie auf der Prädikatenlogik erster Stufe. Neben ihrer Bedeutung als Untersuchungsgegenstand der Mengenlehre findet sie als Grundlagen heute in Teilen der Mathematik, etwa der Kategorientheorie und der algebraischen Geometrie, weite Verwendung. Sie ist nach Alfred Tarski und Alexander Grothendieck benannt.
  • 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. The Mizar system and Metamath use Tarski–Grothendieck set theory for formal verification of proofs.
  • La teoria degli insiemi di Tarski-Grothendieck (TG) è una teoria assiomatica degli insiemi così chiamata in riferimento ai matematici Alfred Tarski e Alexander Grothendieck. Essa è caratterizzata dall'Assioma di Tarski ed è un'estensione non-conservativa della teoria degli insiemi di Zermelo - Fraenkel.
  • A teoria dos conjuntos de Tarski-Grothendieck (TG, assim denominada em referência aos matemáticos Alfred Tarski e Alexander Grothendieck) é uma teoria dos conjuntos axiomática. Também é uma extensão não conservativa da teoria dos conjuntos de Zermelo-Fraenkel (ZFC) e pode ser distinguida de outras teorias dos conjuntos axiomáticas por causa da inclusão do axioma de Tarski, que diz que para cada conjunto existe um universo de Grothendieck a qual pertence. O axioma de Tarski implica na existência de cardinais inacessíveis, fornecendo uma ontologia mais rica do que outras teorias como a ZFC. Por exemplo, adicionar esse axioma dá suporte a teoria das categorias. O sistema Mizar e Metamath usam a teoria dos conjuntos de Tarski-Grothendieck para verificação formal de provas.
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