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.

Although " Amazing Grace " set to " New Britain " was popular, other versions existed regionally.

Although there are no set minimum admissions criteria for Barrett College, the average GPA of incoming freshmen was 3. 84, with average SAT scores of 1314 / 1600 and ACT scores of 29.

Although Rodin is generally considered the progenitor of modern sculpture, he did not set out to rebel against the past.

Although he had for many years " harbored an ambition to create work for a public square ", he " had never set foot in New York, and knew nothing about life in a rapidly evolving metropolis.

Although the Umayyads did not have a historical presence in the region ( no member of the Umayyad family was known to have ever set foot in al-Andalus before ) and there were grave concerns about young Abd al-Rahman's inexperience, several of the lower-ranking Yemenite commanders felt they had little to lose and much to gain, and agreed to support the prince.

Although Zermelo's fix allows a class to describe arbitrary ( possibly " large ") entities, these predicates of the meta-language may have no formal existence ( i. e., as a set ) within the theory.

Although Benedictines do not take a vow of silence, hours of strict silence are set, and at other time silence is maintained as much as is practically possible.

Although each series is set in a different era, all follow the " misfortunes " of Edmund Blackadder ( played by Atkinson ), who in each is a member of a British family dynasty present at many significant periods and places in British history.

Although passengers rarely have cause to refer to either document, all travel is subject to the National Rail Conditions of Carriage and all tickets are valid subject to the rules set out in a number of so-called technical manuals, which are centrally produced for the network.

Although counting the number of elements in a set is a rather broad mathematical problem, many of the problems that arise in applications have a relatively simple combinatorial description.

Although the individual philosophical schools differ considerably, they nevertheless share a common vocabulary and set of concerns.

Although a " bijection " seems a more advanced concept than a number, the usual development of mathematics in terms of set theory defines functions before numbers, as they are based on much simpler sets.

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.

Although Cesare Borgia's father gave him the power to set up, Cesare ruled the Romagna with skill and tact for the most part.

Although Celera was originally set to sequence a composite of DNA samples, partway through the sequencing, Venter switched the samples for his own DNA.

Although these guidelines have been widely implemented, opponents argue that even when a diagnostic criteria set is accepted across different cultures, it does not necessarily indicate that the underlying constructs have any validity within those cultures ; even reliable application can only demonstrate consistency, not legitimacy.

Although rumors of drug use and other " problems on the set " swirled around her dismissal, the producers were adamant that the pregnancy was the only reason her character was written out.

Although various notations are used throughout the literature to denote that two elements a and b of a set are equivalent with respect to an equivalence relation R, the most common are " a ~ b " and " a ≡ b ", which are used when R is the obvious relation being referenced, and variations of " a ~< sub > R </ sub > b ", " a ≡< sub > R </ sub > b ", or " aRb ".

Although he doled out offices, money, and titles mainly to Christian highlanders in the hope of co-opting would-be Eritrean opponents in early 1967, the imperial secret police of Ethiopia also set up a wide network of informants in Eritrea and conducted disappearances, intimidations and assassinations among the same populace driving several prominent political figures into exile.

Although definitions of ecosystem management abound, there is a common set of principles which underlie these definitions.

Although unsure as to why these saints should choose such a " sinner " as their patron, Einhard nonetheless set about ensuring they continued to receive a resting place fitting of their honour.

Although formal certification of his freedom from Burghley's control was deferred until May 1572, Oxford was finally granted the income of £ 666 which his father had intended him to have earlier, but properties set aside to pay his father's debts would not come his way for another decade.

Although individual buses are generally owned by individuals or small companies, the timetables, fares, and levels of service are set by Strandfaraskip Landsins and the government.

Although the wood has been changed to coal, much of it still retains its original cell structure.

Although a straight drum shade would be adequate and sufficiently neutral that the puppets could be changed without disharmony, it is far more fun to create shades in the gay spirit of a child's playtime.

Although England decisively won the Ashes 4 – 1, Bodyline caused such a furore in Australia that diplomats had to intervene to prevent serious harm to Anglo-Australian relations, and the MCC eventually changed the Laws of cricket to curtail the number of leg side fielders.

Although Hole's sound changed over the course of the band's career, the pretty / ugly dynamic has often been noted as a consistent theme in Love's music, most prominently in Hole's first two studio albums.

Although the production techniques have changed, many successful acts since the 1970s have retained the basic disco beat and mentality, and dance clubs have remained popular.

Although Dominican relations with its closest neighbor, the Republic of Haiti, have never been extensive, there are signs this would have changed with the government of President Hipólito Mejía.

Although he was a Roman Catholic, his body was buried in the Basel Minster, despite its having been changed into a Reformed church several years earlier.

Although the family always used the name Eiffel, Gustave's name was registered at birth as Bönickhausen, and was not formally changed to Eiffel until 1880.

Although little changed from the 48-215, marketing campaigns and price cuts kept FJ sales steady until a completely redesigned model was launched.

Although Rosenblum acquiesced to Ben-Gurion's request and legally changed his name to Vardi, he later admitted to regretting not signing as Rosenblum.

Although he won acclaim and major awards for a wide variety of performances, Cinema had changed in the ten years since Walsh last directed Cagney ( in The Strawberry Blonde ), and the actor's portrayal of gangsters had also changed.

Although the vowel was originally written in the English orthography as " oo " ( in spellings such as coola or koolah ), it was changed to " oa " possibly due to an error.

Although the political structure in western Europe had changed, the divide is not as extensive as some historians have claimed., they were only made possible because of the long medieval tradition that established the use of reason as one of the most important of human activities ".

Although the role of the violin in music drastically changed through this period, progress in violin technique was steady but slow.

Although at odds with the scientific establishment for most of his life, Heaviside changed the face of mathematics and science for years to come.

Although Gorbachev's attempts at Perestroika ultimately failed, he drastically changed the perceptions of the outside world towards Russia.

" Although Edmund went out of his way to declare that the story of Father and Son was " scrupulously true ," Thwaite cites a dozen occasions on which either Edmund's " memory betray him — he admitted it was ' like a colander '"— or he " changed things deliberately to make a better story.

Although the Texians originally fought for the reinstatement of the Constitution of 1824, by 1836, the aim of the war had changed.

Although these agreements changed during the 1980s and beyond, it is still expensive to repeat archive television series on British terrestrial television, as new contracts have to be drawn up and payments made to the artists concerned.

Although latterly somewhat changed in nature, this remains primarily an agricultural show.

Although monastics in India traditionally did not work for income, this changed when Buddhism moved to east Asia, so that in China and the surrounding countries monks often engage in agriculture.

Although in rare cases, a court may change the terms of the trust due to unexpected changes in circumstances that make the trust uneconomical or unwieldy to administer, under normal circumstances an irrevocable trust may not be changed by the trustee or the beneficiaries of the trust.

Although the names have changed, the institutions of government remain similar to those that existed before the breakup of the Soviet Union.

Although initially restricted by its constitution to render aid only to nationals from the United Nations ( the Allies ), this was changed late in 1944, in response to pleas from Jewish organizations who were concerned with the fate of surviving Jews of German nationality, to also include " other persons who have been obliged to leave their country or place of origin or former residence or who have been deported therefrom by action of the enemy because of race, religion or activities ' in favor of the United Nations.