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 modern analytical chemistry is dominated by sophisticated instrumentation, the roots of analytical chemistry and some of the principles used in modern instruments are from traditional techniques many of which are still used today.

Although there are few examples of such systems competitive with traditional analysis techniques, potential advantages include size / portability, speed, and cost.

Although chapters 4: 9-10 have said that there is " no king in Zion ", these chapters predict a new military ruler will emerge from Bethlehem, the traditional home of the Davidic monarchy, to restore the security of Israel.

Although this theory has the Persians moving to the strategic offensive, this can be reconciled with the traditional account of the Athenians attacking the Persians by assuming that, seeing the Persians advancing, the Athenians took the tactical offensive, and attacked them.

Although often appearing in stories as male, Coyote can be female, hermaphrodite, or gender changing, in traditional Aboriginal stories.

Although the identification of the Aedicule as the site of Jesus ' tomb is not a tenet of faith for any major Christian denomination, many Catholic and Orthodox Christians hold fast to this traditional location.

Although some Kunama still practice traditional beliefs, most are converts to either Christianity ( Roman Catholic and Protestant ) or Islam.

Although England's first away kits were blue, England's traditional away colours are red shirts, white shorts and red socks.

Although they struggled, these women were eventually vindicated in their bid to reestablish the Church and thereby also to reestablish traditional family life and social stability.

Although Ireland's political spectrum was traditionally divided along Civil War lines, rather than the traditional European left-right spectrum, Fine Gael is described generally as a centre-right, Christian democratic party, with a focus on law and order, enterprise and reward, and fiscal rectitude.

Although the basic idea is recursive, most traditional implementations rearrange the algorithm to avoid explicit recursion.

Although some notable New Testament scholars affirm traditional Johannine scholarship,

Although English largely separates tense and aspect formally, its generally recognized aspects do not correspond very closely to the traditional notion of perfective vs. imperfective aspectual distinction originally devised to classify aspect in most Classical and Slavic languages ( those languages for which the concept of aspect was first proposed in describing non-tense handling of verbal " viewpoint ").

Although many Hellenised Jews seem, as at the Dura-Europos synagogue, to have had images of religious figures, the traditional Mosaic prohibition of " graven images " no doubt retained some effect, although never proclaimed by theologians.

Although traditional Judaism had adopted some features of Kabbalah, it adapted them to fit its own system: it added to its own ritualism the asceticism of the " practical kabbalists " just across the eastern borders in the ancient Greek and Anatolian Jewish communities under the Ottoman Empire, who saw the essence of earthly existence only in fasting, in penance, and in spiritual sadness.

Although not the first Tarzan in movies ( that honor went to Elmo Lincoln ), he was the first to be associated with the now traditional ululating, yodeling Tarzan yell.

Although many municipalities had enacted laws against cross-dressing, some women would socialize in bars as butches: dressed in men's clothing and mirroring traditional masculine behavior.

Although he had never held a traditional military command, Beria made a significant contribution to the victory of the Soviet Union in World War II through his organization of wartime production and his use of partisans.

Although there was no need for a coalition with the Country Party, the traditional non-Labor coalition was retained.

Although it can be used as a traditional networking system, Myrinet is often used directly by programs that " know " about it, thereby bypassing a call into the operating system.

Although these innovations are not all entirely new with Wadzinski, the book marks a significant advancement in published maze puzzles, offering expansions on the traditional puzzles that seem to have been fully informed by various video game innovations and designs, and adds new levels of challenge and complexity in both the design and the goals offered to the puzzle-solver in a printed format.

Although traditional products such as coffee, meat, and sugar continued to lead the list of Nicaraguan exports, the fastest growth is now in nontraditional exports: textile and apparel ; gold ; seafood ; and new agricultural products such as peanuts, sesame, melons, and onions.

Although not specifically Wiccan, one branch of traditional Witchcraft has provided a home for many Neo-Pagan LGBT men and women.

Although none of the major opera houses "... use traditional, Broadway-style sound reinforcement, in which most if not all singers are equipped with radio microphones mixed to a series of unsightly loudspeakers scattered throughout the theatre ", many use a sound reinforcement system for acoustic enhancement, and for subtle boosting of offstage voices, child singers, onstage dialogue, and sound effects ( e. g., church bells in Tosca or thunder effects in Wagnerian operas ).