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...

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

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...

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)...

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

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

applied in the form of the assumption that one can work inside a Grothendieck universe, the two ideas being intimately connected. Suppose that κ {\displaystyle...

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

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

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...

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...

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

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...

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

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

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

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...

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...

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

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

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...

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

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

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

}} (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...

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:...

for theorems about sets alone, and Morse–Kelley set theory and Tarski–Grothendieck set theory, both of which are stronger than ZFC. The above systems can...

dependent type theories. Fibred categories were introduced by Alexander Grothendieck (1959, 1971), and developed in more detail by Jean Giraud (1964, 1971)...