Zorn's lemma, also known as the Kuratowski–Zorn lemma, is a proposition of set theory. It states that a partially ordered set containing upper bounds...

local lemma Nakayama's lemma Poincaré's lemma Riesz's lemma Schur's lemma Schwarz's lemma Sperner's lemma Urysohn's lemma Vitali covering lemma Yoneda's...

of the most common is in the proof that the axiom of choice implies Zorn's lemma. We first prove it for the case where X is chain complete and has no...

Hausdorff maximal principle is an alternate and earlier formulation of Zorn's lemma proved by Felix Hausdorff in 1914 (Moore 1982:168). It states that in...

equivalent to the axiom of choice? A: Zorn's lemon. Individuals not familiar with the mathematical result Zorn's lemma are unlikely to understand the joke...

has a fixed point. This theorem, in turn, can be used to prove that Zorn's lemma is a consequence of the axiom of choice. Algebraic posets Scott topology...

potentially formalisable. Hausdorff maximality theorem Well-ordering theorem Zorn's lemma Axiom of global choice Axiom of countable choice Axiom of dependent choice...

set theory such as the classical theorem that the axiom of choice and Zorn's lemma are equivalent over ZF set theory. The goal of reverse mathematics, however...

Under ZF, the axiom of choice is, in particular, equivalent to (a) Zorn's lemma, (b) Tychonoff's theorem, (c) the weak form of the vector basis theorem...

theorem (to do) Ultrafilter lemma Ultraparallel theorem Urysohn's lemma Van der Waerden's theorem Wilson's theorem Zorn's lemma Bellman–Ford algorithm (to...

spaces Homological algebra Filtration (algebra) Exact sequence Functor Zorn's lemma Semigroup Subsemigroup Free semigroup Green's relations Inverse semigroup...

Axiom of choice Axiom of countable choice Axiom of dependent choice Zorn's lemma Axiom of power set Boolean-valued model Burali-Forti paradox Cantor's...

C {\displaystyle {\mathsf {DC}}} is equivalent to a weakened form of Zorn's lemma; specifically D C {\displaystyle {\mathsf {DC}}} is equivalent to the...

ultrafilter (i.e., a maximal filter), and this is seen by invoking Zorn's lemma. Zorn's lemma is also used to prove Kelley's theorem, that every net has a universal...

existence in a given case is usually a straightforward consequence of Zorn's lemma, based on the idea that a contradiction involves use of only finitely...

theorem, though the term "lemma" is usually kept as part of its name (e.g. Gauss's lemma, Zorn's lemma, and the fundamental lemma). A corollary is a proposition...

choice itself, trichotomy (the ability to compare cardinal numbers), and Zorn's lemma and related maximality principles. This section also includes three more...

equivalent to the axiom of choice. The most important among them are Zorn's lemma and the well-ordering theorem. In fact, Zermelo initially introduced...

Mathematical induction ∈-induction Transfinite number Well-founded induction Zorn's lemma J. Schlöder, Ordinal Arithmetic. Accessed 2022-03-24. It is not necessary...

Axiom of choice Axiom of countable choice Axiom of dependent choice Zorn's lemma Boolean algebra (structure) Boolean-valued model Burali-Forti paradox...

{\displaystyle f(Y)\in Y} . A third version of the axiom, also equivalent, is Zorn's lemma. Since the existence of a choice function when X {\displaystyle X} is...

form an orthomodular lattice. Pseudocomplemented lattice Grätzer (1971), Lemma I.6.1, p. 47. Rutherford (1965), Theorem 9.3 p. 25. Stern, Manfred (1999)...

hypotheses List of inequalities Lists of integrals List of laws List of lemmas List of limits List of logarithmic identities List of mathematical functions...

theory without the axiom of choice or the powerset axiom Zorn 1.  Max Zorn 2.  Zorn's lemma states that if every chain of a non-empty poset has an upper...

ordered by inclusion and the above corollary shows it is non-empty. Zorn's lemma ensures the existence of a maximal element { (Ej, Fj) }. Maximality ensures...

n − 1, n − 2, ..., 2, 1 has length n for any n. The Mostowski collapse lemma implies that set membership is a universal among the extensional well-founded...

set theory such as the classical theorem that the axiom of choice and Zorn's lemma are equivalent over ZF set theory. The goal of reverse mathematics, however...

Virginia Zeani, world-famous operatic soprano Max August Zorn, mathematician and originator of Zorn's lemma David B. Allison, scientist, researcher Martina Arroyo...

Mathematical Society, 42 (6): 366–370, doi:10.1090/S0002-9904-1936-06300-7 Zorn, P. (1986). "The Bieberbach Conjecture" (PDF). Mathematics Magazine. 59 (3):...

for every x, then {ek} is an orthonormal basis. As a consequence of Zorn's lemma, every Hilbert space admits an orthonormal basis; furthermore, any two...