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.

Woodin and his son-in-law Thomas Sanders set up the first general store.

Gruelle's " Raggedy Ann's Sunny Songs " was set to music by William H. Woodin, who in 1934 was appointed FDR's first Secretary of the Treasury.

William Hugh Woodin ( born April 23, 1955 ) is an American mathematician and set theorist at University of California, Berkeley.

Woodin criticizes this view arguing that it leads to a counterintuitive reduction in which all truths in the set theoretical universe can be decided from a small part of it.

If there are infinitely many Woodin cardinals, then projective determinacy holds ; that is, every game whose winning condition is a projective set is determined.

It is also consistent with ZF + DC that every set of reals is Lebesgue measurable ; however, this consistency result, due to Robert M. Solovay, cannot be proved in ZFC itself, but requires a mild large cardinal assumption ( the existence of an inaccessible cardinal ).

Sometimes slightly stronger theories such as Morse-Kelley set theory or set theory with a strongly inaccessible cardinal allowing the use of a Grothendieck universe are used, but in fact most mathematicians can actually prove all they need in systems weaker than ZFC, such as second-order arithmetic.

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.

In 1917 it was established that all cardinals, even cardinal deacons, had to be priests, and in 1962, John XXIII set the norm that all cardinals be bishops.

This definition of cofinality relies on the axiom of choice, as it uses the fact that every non-empty set of cardinal numbers has a least member.

Later modifications to his theory allowed for an additional set of eight " secondary Cardinal Vowels " with reverse lip shapes, permitting the representation of eight secondary cardinal vowels ( front rounded and back unrounded ).

Formal-ontological categories relate objects and include notions such as set, cardinal number, ordinal number, part and whole, relation, and so on.

Barry Mitchell has shown ( in " The cohomological dimension of a directed set ") that if I has cardinality ( the dth infinite cardinal ), then R < sup > n </ sup > lim is zero for all n ≥ d + 2.

Now is the cardinality of the set of countable ordinals, and this is merely a special and the simplest way of generating a higher cardinal.

The Principia covered only set theory, cardinal numbers, ordinal numbers, and real numbers.

The uncountability of a set is closely related to its cardinal number: a set is uncountable if its cardinal number is larger than that of the set of all natural numbers.

The size ( number of elements, cardinal number ) of a finite set is equal to the order type.

In 1651 when Mazarin had been forced into exile, the Prince was for a time brought onto the conseil du roi, and an ( admittedly very hostile ) contemporary the duchesse de Nemours described him as a ' prime minister without being aware of it '; there were suggestions that Mazarin's opponents within the court had raised him up as a rival to the cardinal with the Queen, but this is unlikely, especially since Mazarin himself urged the Queen to follow Thomas ' advice, and it is more probable that Mazarin backed the Prince as someone who would keep other rivals from gaining control in his absence but who would never have the status within France to set himself up as a permanent replacement for the Cardinal.

In 1651 when Mazarin had been forced into exile, the Prince was for a time brought onto the conseil du roi, and an ( admittedly very hostile ) contemporary the duchesse de Nemours described him as a ' prime minister without being aware of it '; there were suggestions that Mazarin's opponents within the court had raised him up as a rival to the cardinal with the Queen, but this is unlikely, especially since Mazarin himself urged the Queen to follow Thomas ' advice, and it is more probable that Mazarin backed the Prince as someone who would keep other rivals from gaining control in his absence but who would never have the status within France to set himself up as a permanent replacement for the Cardinal.

* Continuum ( set theory ), the real line or the corresponding cardinal number

* Cardinality of the continuum, a cardinal number that represents the size of the set of real numbers

Sculptures embodying seasons and cardinal points are set above the tower's three tiers.

Billy decided to set an example by arresting one of the ranchers, named Ed Dunn, who lived at Rock Fort.

* In the year 1000, the Icelander Leif Ericson was the first European to set foot on North American soil, corresponding to today's Eastern coast of Canada, i. e. the province of Newfoundland and Labrador, including the area of land named " Vinland " by Ericson.

Its notoriety arises from an incident in 1894 in which the then owner, an English landlady named Agnes McDonnell, was savagely beaten and the house set alight, allegedly by a local man, James Lynchehaun.

In 2010, Cuarón began to develop a science fiction film named Gravity, which he is set to direct.

A publisher named Edwin Othello Excell gave the version of " Amazing Grace " set to " New Britain " immense popularity by publishing it in a series of hymnals that were used in urban churches.

With their consent, he set up a rival emperor, the prefect of the city, a Greek named Priscus Attalus.

As nothing cannot be known by any means or method it must mean, in the context of the question, that a specific named object is present or not present in the observer's experience of a set of objects, conditions for which English uses " is " or " is not.

The show is set in the Cheers bar ( named for the toast " Cheers ") in Boston, Massachusetts, where a group of locals meet to drink, relax, chat and have fun.

In Coleman, Texas, a farmer named Reggie Lagow caught an animal in a trap he set up after the deaths of a number of his chickens and turkeys.

In his 16-year career in the NFL, Anderson made four trips to the Pro Bowl, won four passing titles, was named NFL MVP in 1981, and set the record for completion percentage in a single season in 1982 ( 70. 66 %).

Independently and contemporaneously, a 19-year-old American student named Robert Lee Moore published an equivalent set of axioms.

Revolving around an Englishman living in 1930s London named Robert Denvers who has recollections of a previous life as a Bronze Age Cypriot – an allusion to Gardner himself – the primary plot of A Goddess Arrives is set in ancient Cyprus and featured a queen, Dayonis, who practices sorcery in an attempt to help her people defend themselves from invading Egyptians.

The Jupiter Ace was a British home computer of the early 1980s, produced by a company, set up for the purpose, named Jupiter Cantab.

In 1917 a group of secular nationalists called the Alash Orda Horde of Alash, named for a legendary founder of the Kazakh people, attempted to set up an independent national government – the Alash Autonomy.

Baldwin brought with him an Armenian wife, traditionally named Arda ( although never named such by contemporaries ), whom he had married to gain political support from the Armenian population in Edessa, and whom he quickly set aside when he no longer needed Armenian support in Jerusalem.

* Desolation Road ( 1988 ) by Ian McDonald is a magic-realist science fiction novel set on a planet that's never explicitly named ( though the name " Ares " makes frequent appearances in various contexts ) but is clearly meant to be a terraformed Mars.

If the accusers were among those named, the defendant was set free and the charges dismissed ; the accusers would face life imprisonment.

Also, the deepest-diving underwater ROV, which recently set a record for exploring the Challenger Deep of the Mariana Trench, is named after this figure.

In his telling, the female pope is not named, and the events are set in 1099.

In 1780, a farm was reclaimed on the site of an ancient tenement near the Two Bridges, and in 1785, Sir Thomas Tyrwhitt set about improving the moor at a place which he named Tor Royal, about south-east of Princetown.

At the end of nine months he was set free by the clergy ; but a matron named Lucina having had her house on the Via Lata consecrated by him as " titulus Marcelli " he was again condemned to the work of attending to the horses brought into the station, in which menial occupation he died.

The film Paris, Texas by Wim Wenders was named after the city, but was not set there.

Perth is set on the Swan River, named after the native black swans in 1697 by Willem de Vlamingh, captain of a Dutch expedition and namer of WA's Rottnest Island.

Munroe's method of making pencils was painstakingly slow, and in the neighbouring town of Acton, a pencil mill owner named Ebenezer Wood set out to automate the process at his own pencil mill located at Nashoba Brook along the old Davis Road.