In mathematics, a Grothendieck universe is a set U with the following properties: If x is an element of U and if y is an element of x, then y is also...

to universes which is historically connected with category theory. This is the idea of a Grothendieck universe. Roughly speaking, a Grothendieck universe...

Tarski–Grothendieck set theory (TG, named after mathematicians Alfred Tarski and Alexander Grothendieck) is an axiomatic set theory. It is a non-conservative...

contradictions, and any Grothendieck universe satisfies the new pair of properties. However, whether Grothendieck universes exist is a question beyond...

{\displaystyle V_{\kappa }} is a Grothendieck universe. Conversely, if U {\displaystyle U} is a Grothendieck universe then there is a strongly inaccessible...

contraposition to the distinguished sets that are elements of a Grothendieck universe. The most popular axiomatic set theories, Zermelo–Fraenkel set theory...

Grothendieck trace formula Grothendieck trace theorem Grothendieck pretopology Grothendieck topoi Grothendieck topology Grothendieck universe Institut Montpelliérain...

Another solution is to assume the existence of Grothendieck universes. Roughly speaking, a Grothendieck universe is a set which is itself a model of ZF(C)...

of Fermat's Last Theorem implicitly relies on the existence of Grothendieck universes, very large infinite sets, for solving a long-standing problem that...

integers whose sum of reciprocals converges Small set, an element of a Grothendieck universe Ideal (set theory) Natural density Large set (disambiguation) This...

is assumed, then the sets of smaller rank form a model of ZF (a Grothendieck universe), and its subsets can be thought of as "classes". In ZF, the concept...

such as Grothendieck universes, there exist both sets that belong to the universe, called “small sets” and sets that do not, such as the universe itself...

not contain itself, because it is not itself a set. Universe (mathematics) Grothendieck universe Domain of discourse Von Neumann–Bernays–Gödel set theory...

theory with a strongly inaccessible cardinal allowing the use of a Grothendieck universe is used, but in fact, most mathematicians can actually prove all...

problem: One can work with Grothendieck universes: a stack is then a functor between classes of some fixed Grothendieck universe, so these classes and the...

which is used in Tarski–Grothendieck set theory and states (in the vernacular) that every set belongs to some Grothendieck universe, is stronger than the...

metamath have adopted Tarski–Grothendieck set theory, an extension of ZFC, so that proofs involving Grothendieck universes (encountered in category theory...

One can obtain a true field by limiting the construction to a Grothendieck universe, yielding a set with the cardinality of some strongly inaccessible...

relation defined on proper classes, as an alternative to postulating a Grothendieck universe; it may also be used as an alternative to choice in the proof of...

In mathematics, in set theory, the constructible universe (or Gödel's constructible universe), denoted by L , {\displaystyle L,} is a particular class...

of sets on the site of compact Hausdorff spaces (with some fixed Grothendieck universes). The notion was introduced by Barwick and Haine to provide a convenient...

Wiles's proof of Fermat's Last Theorem, which relies implicitly on Grothendieck universes, whose existence requires the addition of a new axiom to set theory...

setting, working with concepts such as Grothendieck toposes and Grothendieck universes. With hindsight, much of this machinery proved unnecessary for most...

set-theoretic in nature: pyknotic theory depends on a choice of Grothendieck universes, whereas condensed mathematics can be developed strictly within...

greatest editorial failures of all time." French mathematician Alexandre Grothendieck wrote about The Sleepwalkers that "The metaphor of the 'sleepwalker'...

theory or Tarski–Grothendieck set theory, albeit that in very many cases the use of large cardinal axioms or Grothendieck universes is formally eliminable...

and C {\displaystyle C} an ∞-category (a weak Kan complex). Fix a Grothendieck universe. Then, roughly, a limit of a functor f : I → C {\displaystyle f:I\to...

is the notion of a small set; i.e., one has made a choice of a Grothendieck universe. Kashiwara & Schapira 2006, Corollary 2.4.3. Kashiwara & Schapira...

Grothendieck Grothendieck universes for sets as part of foundations for categories 1972 Jean Bénabou–Ross Street Cosmoses which categorize universes:...

A^{c}} (or A′), is the set of elements not in A. When all elements in the universe, i.e. all elements under consideration, are considered to be members of...