Definitions for zorn's lemma
zorn's lem·ma

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1. Zorn's lemma

   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 for every chain (that is, every totally ordered subset) necessarily contains at least one maximal element. The lemma was proved (assuming the axiom of choice) by Kazimierz Kuratowski in 1922 and independently by Max Zorn in 1935. It occurs in the proofs of several theorems of crucial importance, for instance the Hahn–Banach theorem in functional analysis, the theorem that every vector space has a basis, Tychonoff's theorem in topology stating that every product of compact spaces is compact, and the theorems in abstract algebra that in a ring with identity every proper ideal is contained in a maximal ideal and that every field has an algebraic closure.Zorn's lemma is equivalent to the well-ordering theorem and also to the axiom of choice, in the sense that within ZF (Zermelo–Fraenkel set theory without the axiom of choice) any one of the three is sufficient to prove the other two. An earlier formulation of Zorn's lemma is Hausdorff's maximum principle which states that every totally ordered subset of a given partially ordered set is contained in a maximal totally ordered subset of that partially ordered set.

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1. Zorn's lemma

   Zorn's lemma, also known as the Kuratowski–Zorn lemma, is a proposition of set theory that states: Suppose a partially ordered set P has the property that every chain has an upper bound in P. Then the set P contains at least one maximal element. It is named after the mathematicians Max Zorn and Kazimierz Kuratowski. The terms are defined as follows. Suppose is a partially ordered set. A subset T is totally ordered if for any s, t in T we have s ≤ t or t ≤ s. Such a set T has an upper bound u in P if t ≤ u for all t in T. Note that u is an element of P but need not be an element of T. An element m of P is called a maximal element if there is no element x in P for which m < x. Note that P is not required to be non-empty. However, the empty set is a chain, hence is required to have an upper bound, thus exhibiting at least one element of P. An equivalent formulation of the lemma is therefore: Suppose a non-empty partially ordered set P has the property that every non-empty chain has an upper bound in P. Then the set P contains at least one maximal element. The distinction may seem subtle, but proofs involving Zorn's lemma often involve taking a union of some sort to produce an upper bound. The case of an empty chain, hence empty union is a boundary case that is easily overlooked.

How to pronounce zorn's lemma?

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Numerology

1. Chaldean Numerology

   The numerical value of zorn's lemma in Chaldean Numerology is: 5

2. Pythagorean Numerology

   The numerical value of zorn's lemma in Pythagorean Numerology is: 1

References

1. ^ Wikipedia
   https://en.wikipedia.org/wiki/Zorn'S_Lemma
2. ^ Wikidata
   https://www.wikidata.org/w/index.php?search=zorn's lemma