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 my shot killed his horse, he rolled off the bale on top of me.

Although New Orleans was not to learn of it for a spell, she also was a sadist, a nymphomaniac and unobtrusively mad -- the perpetrator of some of the worst crimes against humanity ever committed on American soil.

Although the impact of these increases on our industry's shipments will be gradual, on balance I look for another good increase in shipments in the coming year, to at least $6 billion.

Although faculties insist on governing themselves, they grant little prestige to a member who actively participates in college or university government.

-- Although there was some variation in results which must be attributed either to trace impurities or to variation in wall effects, the photochemical exchange in the gas phase was sufficiently reproducible so that it seemed meaningful to compare the reaction rates in different series of reaction tubes for the purpose of obtaining information on the effect of chlorine concentration and of carbon tetrachloride concentration on the reaction rate.

Although the standard deviation values on which spread of the lines is based are relatively larger for those centers which begin to ossify early ( Table 1 ), there are considerable differences in this value between centers having the closely timed Onsets.

Although no drugs act exclusively on the hypothalamus or a part of it, there is sufficient specificity to distinguish drugs which shift the hypothalamic balance to the sympathetic side from those which produce a parasympathetic dominance.

Although admitting Brown's guilt on legal grounds, Day said that, `` Brown is no common criminal ; ;

Although the first polyether foams on the market had to be produced by the two-step prepolymer method, today, thanks to new catalysts, they can be produced by a one-shot technique.

Although she weighed only 108 pounds when she visited him, Carroll permitted her to go on a 10-day fast in which she took nothing but water.

Although he pointed out that mandatory legislation impinging on home rule is basically distasteful, he added that the vital interest in election results transcended county lines.

Although modern scholars have expressed surprise that `` the simple magic square of three '', a mere `` mathematical puzzle '', was able to exert a considerable influence on the minds and imaginations of the cultured Chinese for so many centuries, they could have found most of the answers right within the square itself.

Although I absolutely reject the Platonism of it, I have literally squealed with delight at the imperturbable perfection with which the position is laid down on page after page ''.

Although a skillful politician and a courageous and honest man, Mr. Nixon, Mr. White believes, ignored his own top-level planners, wasted time and effort in the wrong regions, missed opportunities through indecision and damaged his chances on television.

Although they " were expecting to see activity in the brain's reward centers ", based on the idea that " people perform altruistic acts because they feel good about it ", what they found was that " another part of the brain was also involved, and it was quite sensitive to the difference between doing something for personal gain and doing it for someone else's gain ".

Although digital setting circles can be used to display a telescope's RA and Dec coordinates, they are not simply a digital read-out of what can be seen on the telescope's analog setting circles.

Although the Indian and Greek concepts of the atom were based purely on philosophy, modern science has retained the name coined by Democritus.

Although single genes have little effect on complex traits and interact heavily both between themselves and with the external factors, research is underway to unravel possible molecular mechanisms underlying anxiety and comorbid conditions.

Although originally considered a subspecies of A. ramidus, in 2004 anthropologists Yohannes Haile-Selassie, Gen Suwa, and Tim D. White published an article elevating A. kadabba to species level on the basis of newly-discovered teeth from Ethiopia.

Although he was committed to maintaining what the church had defined at Nicaea, Constantine was also bent on pacifying the situation and eventually became more lenient toward those condemned and exiled at the council.

Although much of the church hierarchy in the East had opposed the Nicene creed in the decades leading up to Theodosius ' accession, he managed to achieve unity on the basis of the Nicene creed.

Although Abensberg was an autonomous city, it remained dependent on the powerful Dukes of Bavaria.

Although Dirac did not himself use the term antimatter, its use follows on naturally enough from antielectrons, antiprotons, etc.