Large cardinals | Category theory | Set-theoretic universes
In mathematics, a Grothendieck universe is a set U with the following properties: 1. * If x is an element of U and if y is an element of x, then y is also an element of U. (U is a transitive set.) 2. * If x and y are both elements of U, then is an element of U. 3. * If x is an element of U, then P(x), the power set of x, is also an element of U. 4. * If is a family of elements of U, and if I is an element of U, then the union is an element of U. A Grothendieck universe is meant to provide a set in which all of mathematics can be performed. (In fact, uncountable Grothendieck universes provide models of set theory with the natural ∈-relation, natural powerset operation etc.). Elements of a Grothendieck universe are sometimes called small sets. The idea of universes is due to Alexander Grothendieck, who used them as a way of avoiding proper classes in algebraic geometry. The existence of a nontrivial Grothendieck universe goes beyond the usual axioms of Zermelo–Fraenkel set theory; in particular it would imply the existence of strongly inaccessible cardinals.Tarski–Grothendieck set theory is an axiomatic treatment of set theory, used in some automatic proof systems, in which every set belongs to a Grothendieck universe.The concept of a Grothendieck universe can also be defined in a topos. (Wikipedia).
Céline Pessis - L'engagement d'Alexandre Grothendieck durant la première moitié des années 1970
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From playlist ICTS Colloquia
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Abonniert den Kanal, damit er auch in Zukunft bestehen kann. Es ist vollkommen kostenlos und ihr werdet direkt informiert, wenn ich einen Livestream anbiete. Hier erzähle ich etwas über die Konstruktion der Grothendieck-Gruppe in einem abstrakten Rahmen, aber mit elementaren Rechnungen. E
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Signature de la convention de création du "Laboratoire Alexander Grothendieck"
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Yonatan Harpaz - New perspectives in hermitian K-theory III
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From playlist New perspectives on K- and L-theory