The most important among them are Zorn's lemma and the well-ordering theorem.

** Zorn's lemma: Every non-empty partially ordered set in which every chain ( i. e. totally ordered subset ) has an upper bound contains at least one maximal element.

Using Zorn's lemma, it can be shown that every field has an algebraic closure, and that the algebraic closure of a field K is unique up to an isomorphism that fixes every member of K. Because of this essential uniqueness, we often speak of the algebraic closure of K, rather than an algebraic closure of K.

In mathematics, the Hausdorff maximal principle is an alternate and earlier formulation of Zorn's lemma proved by Felix Hausdorff in 1914 ( Moore 1982: 168 ).

The proof that the Hausdorff maximal principle is equivalent to Zorn's lemma is very similar to this proof.

The extension ψ is in general not uniquely specified by φ, and the proof gives no explicit method as to how to find ψ: in the case of an infinite dimensional space V, it depends on Zorn's lemma, one formulation of the axiom of choice.

A good stepping stone leads to many others, so some of the most powerful results in mathematics are known as lemmata, such as Bézout's lemma, Urysohn's lemma, Dehn's lemma, Euclid's lemma, Farkas ' lemma, Fatou's lemma, Gauss's lemma, Nakayama's lemma, Poincaré's lemma, Riesz's lemma, Schwarz's lemma, Itō's lemma and Zorn's lemma.

He is best known for Zorn's lemma, a powerful tool in set theory that is applicable to a wide range of mathematical constructs such as vector spaces, ordered sets, etc.

* Zorn's lemma

Examples include Gauss's lemma and Zorn's lemma.

# REDIRECT Zorn's lemma

Composer John Zorn's Redbird ( 1995 ) was inspired by and dedicated to Martin.

Zorngården remains today much as it was at the time of Emma Zorn's death in 1942.

Robert Zorn's 2012 book, Cemetery John, proposes that Hauptmann was the foot soldier in a conspiracy with two other German-born men, John and Walter Knoll.

The reverse mathematics program was foreshadowed by results in set theory such as the classical theorem that the axiom of choice and Zorn's lemma are equivalent over ZF set theory.

During Zorn's 50th Birthday Celebration at Tonic in September 2003, Electric Masada was recorded live.

It is shown today as it was at the time of Emma Zorn's death in 1942.

Their double-CD album, Decided ... Already the Motionless Heart of Tranquility, Tangling the Prayer Called " I ", was released on John Zorn's Tzadik Records label in March 1999.

Their 1993 album Allegorical Misunderstanding was released on John Zorn's record label, Avant, although most of their albums have come out on independent label P. S. F.

After a few months of practicing together, the three met Brad Breeck, who was studying at the California Institute of Arts with Buchla and had performed versions of John Zorn's " strategy game piece " Cobra in an ensemble led by Buchla.

* Masada: Alef the first album by John Zorn's Masada

They have also contributed to many compilation CDs, including the first song on John Zorn's Serge Gainsbourg tribute album.

In addition to contributing to many of John Zorn's produced compilation records, including one of Marc Bolan and the first track on the Serge Gainsbourg tribute, Charles and Bloedow also released two complete albums of Sephardic songs on Zorn's Tzadik label, the first being " La Mar Enfortuna " and the most recent " Convivencia ".

* In May 2006, he wrote a column coining " Zorn's Law ", stating that, in any debate, the first person to hurl the insult, " get a life!

The Allmusic review by Marc Gilman awarded the album 4½ stars noting that " While some compositions retain their original structure and sound, some are expanded and probed by Zorn's arrangements, and resemble avant-garde classical music more than jazz.

For instance, the third proof uses that every filter is contained in an ultrafilter ( i. e., a maximal filter ), and this is seen by invoking Zorn's lemma.

* John Zorn's Spillane is a tribute to Mickey Spillane, which uses quotes of Mike Hammer voiced by John Lurie.

Ribot has appeared on numerous recordings by John Zorn, including many of Zorn's Filmworks recordings, solo performances on Zorn's Masada Guitars ( also featuring Bill Frisell and Tim Sparks ), and is a member of Zorn's Bar Kokhba Sextet and Electric Masada.

Records released under his own name include Cathode ( 1999 ), which includes pieces that are sine wave-based or made from samples, and Anode ( 2001 ), a group improvisation where the players are constrained by certain pre-determined rules ( similar in some respects to John Zorn's game pieces ).

The performing style of both groups is characterised by the use of improvisation ( sometimes conducted by Zorn himself ) and its use of the inflections of Jewish music that are part of the compositional language of Zorn's " Masada " themes.

New York City Opera's 2010 – 2011 season included a new production of Leonard Bernstein's A Quiet Place directed by Christopher Alden ; Richard Strauss's Intermezzo directed by Leon Major ; and a new production titled Monodramas which consisted of three solo one-act works: John Zorn's La Machine de l ’ être, Arnold Schoenberg's Erwartung, and Morton Feldman's Neither.

They were heavily influenced by so-called no wave music ( Arto Lindsay had played in the seminal no-wave band DNA ), but their music also contained elements of funk and of the improvisational jazz stylings that would become Zorn's trademark.

* Masada ( band ), a jazz quartet fronted by avant-garde composer John Zorn, and a series of Zorn's musical projects based on compositions originally written for this quartet, including:

# Tukey's lemma: In, partially ordered by inclusion, the union of every chain of elements of also belong to, therefore, by Zorn's lemma, contains at least one maximal element.

Tenshi No Gijinka is an album by Japanese musician Keiji Haino, released on John Zorn's Tzadik label in 1995.

Kuratowski proved the Kuratowski-Zorn lemma ( often called just Zorn's lemma ) in 1922 ( Fundamenta Mathematicae, vol. 3 ).

Zorn's lemma, also known as the Kuratowski – Zorn lemma, is a proposition of set theory that states:

Zorn's lemma states that, if a poset has an upper bound for every chain, then it has a maximal element.

Suppose − 1 is not a sum of squares, then a Zorn's Lemma argument shows that the prepositive cone of sums of squares can be extended to a positive cone P ⊂ F.

Max Zorn's grandson Eric Zorn is a columnist for the Chicago Tribune.

The main part of Zorngården consists of Zorn's home and a museum with his art, but there are two other museums that also are part of the Zorn Collection.

Zorn's father, economist Eugene Zorn, had been investigating an incident from his teen years that convinced him he had witnessed the conspiracy being discussed.

The Zorn Collections is the administrator of Anders and Emma Zorn's donation to the Swedish State.

The Zorn Collections consists of four museums: Zorngården, Zorn Museum, Zorn's Gammelgård & Textilkammare and Zorn's Gopsmor.

Beyond her solo recordings, she has recorded or performed with Dave Douglas, Butch Morris, Kim Gordon, Thurston Moore, and many others, including as Hemophiliac, a trio with John Zorn and singer Mike Patton, as well as being a member of Zorn's Electric Masada.

The often rocky relationship Campbell had with head coach Jim Zorn came to an end with Zorn's firing.

Eric Zorn's wife, Johanna Zorn, is founder and executive director of the Third Coast International Audio Festival on Chicago Public Radio.