The Axioms required to make the theoretical machinery operate are set out tersely and powerfully, so that all permissible operations within the theory can be traced rigorously back to these axioms, rules, and primitive notions.

The strongest appeal of the Copernican formulation consisted in just this: ideally, the justification for dealing with special problems in particular ways is completely set out in the basic ' rules ' of the theory.

In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that " the product of a collection of non-empty sets is non-empty ".

Although originally controversial, the axiom of choice is now used without reservation by most mathematicians, and it is included in ZFC, the standard form of axiomatic set theory.

* ZF – Zermelo – Fraenkel set theory omitting the Axiom of Choice.

* ZFC – Zermelo – Fraenkel set theory, extended to include the Axiom of Choice.

These are equivalent in the sense that, in the presence of other basic axioms of set theory, they imply the axiom of choice and are implied by it.

However, that particular case is a theorem of Zermelo – Fraenkel set theory without the axiom of choice ( ZF ); it is easily proved by mathematical induction.

For finite sets X, the axiom of choice follows from the other axioms of set theory.

A proof requiring the axiom of choice may establish the existence of an object without explicitly defining the object in the language of set theory.

For example, while the axiom of choice implies that there is a well-ordering of the real numbers, there are models of set theory with the axiom of choice in which no well-ordering of the reals is definable.

It is possible to prove many theorems using neither the axiom of choice nor its negation ; such statements will be true in any model of Zermelo – Fraenkel set theory ( ZF ), regardless of the truth or falsity of the axiom of choice in that particular model.

In constructive set theory, however, Diaconescu's theorem shows that the axiom of choice implies the law of the excluded middle ( unlike in Martin-Löf type theory, where it does not ).

Thus the axiom of choice is not generally available in constructive set theory.

A cause for this difference is that the axiom of choice in type theory does not have the extensionality properties that the axiom of choice in constructive set theory does.

Some results in constructive set theory use the axiom of countable choice or the axiom of dependent choice, which do not imply the law of the excluded middle in constructive set theory.

Because of independence, the decision whether to use of the axiom of choice ( or its negation ) in a proof cannot be made by appeal to other axioms of set theory.

In class theories such as Von Neumann – Bernays – Gödel set theory and Morse – Kelley set theory, there is a possible axiom called the axiom of global choice which is stronger than the axiom of choice for sets because it also applies to proper classes.

In this sense almost all reals are not a member of the Cantor set even though the Cantor set is uncountable.

Some believe that Georg Cantor's set theory was not actually implicated by these paradoxes ( see Frápolli 1991 ); one difficulty in determining this with certainty is that Cantor did not provide an axiomatization of his system.

However, the term naive set theory is also used in some literature to refer to the set theories studied by Frege and Cantor, rather than to the informal counterparts of modern axiomatic set theory ; care is required to tell which sense is intended.

For example, Georg Cantor ( who introduced this concept ) demonstrated that the real numbers cannot be put into one-to-one correspondence with the natural numbers ( non-negative integers ), and therefore that the set of real numbers has a greater cardinality than the set of natural numbers.

* The Cantor set is compact.

In fact, every compact metric space is a continuous image of the Cantor set.

* Since the p-adic integers are homeomorphic to the Cantor set, they form a compact set.

In mathematics, the Cantor set is a set of points lying on a single line segment that has a number of remarkable and deep properties.

Although Cantor himself defined the set in a general, abstract way, the most common modern construction is the Cantor ternary set, built by removing the middle thirds of a line segment.

Cantor himself only mentioned the ternary construction in passing, as an example of a more general idea, that of a perfect set that is nowhere dense.

The Cantor ternary set is created by repeatedly deleting the open middle thirds of a set of line segments.

A fundamental theorem due to Georg Cantor shows that it is possible for infinite sets to have different cardinalities, and in particular the set of real numbers and the set of natural numbers do not have the same cardinal number.

Basic theories, such as arithmetic, real analysis and complex analysis are often introduced non-axiomatically, but implicitly or explicitly there is generally an assumption that the axioms being used are the axioms of Zermelo – Fraenkel set theory with choice, abbreviated ZFC, or some very similar system of axiomatic set theory like Von Neumann – Bernays – Gödel set theory, a conservative extension of ZFC.

* 1915 – Tadeusz Kantor, Polish painter, set designer and theater director ( d. 1990 )

* 1869 – Hans Poelzig, German architect, painter, and set designer ( d. 1936 )

* 1536 – Buddhist monks from Kyoto, Japan's Enryaku-ji temple set fire to 21 Nichiren temples throughout in what will be known as the Tenbun Hokke Disturbance.

In 1948 Australia set new standards, completely outplaying their hosts to win 4 – 0 with one draw.

; Amber mutations: were the first set of nonsense mutations to be discovered, isolated by graduate student Harris Bernstein in experiments designed to resolve a debate between Richard Epstein and Charles Steinberg.

Bernstein suggested they rework East Side Story and set it in Los Angeles, but Laurents felt he was more familiar with Puerto Ricans and Harlem than he was with Mexican Americans and Olvera Street.

With the help of Oscar Hammerstein, Laurents convinced Bernstein and Sondheim to move " One Hand, One Heart ", which he considered too pristine for the balcony scene, to the scene set in the bridal shop, and as a result " Tonight " was written to replace it.

He states: " And yet, even the elementary form that Russell < sup > 9 </ sup > gave to the set-theoretic antinomies could have persuaded them König, Jourdain, F. Bernstein that the solution of these difficulties is not to be sought in the surrender of well-ordering but only in a suitable restriction of the notion of set ".

* Bernstein v. United States, a set of court cases brought by Daniel J. Bernstein challenging restrictions on the export of encryption software outside the United States

Bernstein v. United States is a set of court cases brought by Daniel J. Bernstein challenging restrictions on the export of cryptography from the United States.

* Cantor – Bernstein – Schroeder theorem, a mathematical theorem in set theory

On the Town is a musical with music by Leonard Bernstein and book and lyrics by Betty Comden and Adolph Green, based on Jerome Robbins ' idea for his 1944 ballet Fancy Free, which he had set to Bernstein's music.

The Jerome Robbins ballet " Fancy Free " ( 1944 ), with music by Leonard Bernstein, was a hit for the American Ballet Theatre, and Oliver Smith ( the set designer ) and his business partner, Paul Feigay, thought that the ballet could be turned into a Broadway musical.

" Oliver Smith, set designer and collaborator on Fancy Free, knew Leonard Bernstein and eventually Robbins and Bernstein met to work on the music.

The second thread is set in the University of California, San Diego, in La Jolla, California, in 1962 where a young scientist, Gordon Bernstein, discovers anomalous noise in a physics experiment relating to spontaneous resonance and indium antimonide.

As the Kyrie is the first item in settings of the mass ordinary and the second in the requiem mass ( the only mass proper set regularly over the centuries ), numerous composers have included Kyries in their masses, including Guillaume de Machaut, Guillaume Dufay, Johannes Ockeghem, Josquin des Prez, Giovanni Pierluigi da Palestrina, Johann Sebastian Bach, Joseph Haydn, Wolfgang Amadeus Mozart, Luigi Cherubini, Franz Schubert, Ludwig van Beethoven, Gabriel Fauré, Hector Berlioz, Charles Gounod, Giuseppe Verdi, Ralph Vaughan Williams, Igor Stravinsky, Leonard Bernstein, Benjamin Britten, Arvo Pärt, Mark Alburger, and Erling Wold.

A generalization of the concept of a transversal would be a set that just has a non-empty intersection with each member of C. An example of this would be a Bernstein set, which is defined as a set that has a non-empty intersection with each set of C, but contains no set of C, where C is the collection of all perfect sets of a topological Polish space.

In the 1950s, coinciding with the growth in television, the Granada Television franchise was set up by Sidney Bernstein.

Nicomide is distributed by Sirius Laboratories Inc., founded by Dr. Joel E. Bernstein, and now a wholly owned subsidiary of DUSA Pharmaceuticals, Inc. DUSA Pharmaceuticals ceased distributing Nicomide in June 2008 and set up an agreement with River's Edge Pharmaceuticals to allow them to distribute the product under DSHEA in August 2008 but the deal appears to have fallen through.