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 amaranth was cultivated on a large scale in ancient Mexico, Guatemala, and Peru, nowadays it is only cultivated on a small scale there, along with India, China, Nepal, and other tropical countries ; thus, there is potential for further cultivation in those countries, as well as in the U. S. In a 1977 article in Science, amaranth was described as " the crop of the future.

Although most cultural systems of astrology share common roots in ancient philosophies that influenced each other, many have unique methodologies which differ from those developed in the West.

Although the various Baltic tribes were mentioned by ancient historians as early as 98 B. C., the first attestation of a Baltic language was in about 1350, with the creation of the Elbing Prussian Vocabulary, a German to Prussian translation dictionary.

Although he must have been familiar with the ancient name, Balcia, meaning a supposed island in the Baltic Sea, and although he may have been aware of the Baltic words containing the stem balt -, " white ", as " swamp ", he reports that he followed the local use of balticus from baelt (" belt ") because the sea stretches to the east " in modum baltei " (" in the manner of a belt ").

Although traditionally focused on ancient Greece and Rome, the study now encompasses the entire ancient Mediterranean world, thus expanding their studies to Northern Africa and parts of the Middle East.

Although it has sometimes been claimed that ancient Egypt lacked urbanism, several types of urban settlements were found in ancient times.

Although Confucius claimed that he never invented anything but was only transmitting ancient knowledge ( see Analects VII, 1 ), he did produce a number of new ideas.

Although much of Chinese philosophy begins in the Warring States period, elements of Chinese philosophy have existed for several thousand years ; some can be found in the Yi Jing ( the Book of Changes ), an ancient compendium of divination, which dates back to at least 672 BCE.

Although the People's Republic of China has been historically hostile to the philosophy of ancient China, the influences of past are still deeply ingrained in the Chinese culture.

Although Tiberius was 77 and on his death bed, some ancient historians still conjecture that he was murdered.

Although the catapult has been used since ancient times, it has proven to be one of the most effective mechanisms during warfare.

Although not part of the modern pentathlon, it was one of the events of the ancient pentathlon, which can be dated at least back to 708 BC.

Although ancient Greek philosophers believed in individual virtue ethics, philosophers like " Plato, Aristotle, and the Stoics not accept the formal principle that whatever the good is, we should seek only our own good, or prefer it to the good of others.

Although most of the relevant tissues and endocrine glands had been identified by early anatomists, a more humoral approach to understanding biological function and disease was favoured by the ancient Greek and Roman thinkers such as Aristotle, Hippocrates, Lucretius, Celsus, and Galen, according to Freeman et al., and these theories held sway until the advent of germ theory, physiology, and organ basis of pathology in the 19th century.

Although mythical elements are not so prominent in Exodus as in Genesis, the echoes of ancient legends are crucial to understanding the book's origins and purpose: for example, the story of the infant Moses's salvation from the Nile has its basis in an earlier legend of king Sargon, while the story of the parting of the Red Sea trades on Mesopotamian creation mythology.

Although the acronym ECU is formed from English words, écu is also the name of an ancient French coin.

Although Baltic Germans at large regarded the future of Estonians as being a fusion with themselves, the Estophile educated class admired the ancient culture of the Estonians and their era of freedom before the conquests by Danes and Germans in the 13th century.

Although there is no record to establish when plants were first used for medicinal purposes ( herbalism ), the use of plants as healing agents is an ancient practice.

Although in later times Hecate's dog came to be thought of as a manifestation of restless souls or demons who accompanied her, its docile appearance and its accompaniment of a Hecate who looks completely friendly in many pieces of ancient art suggests that its original signification was positive and thus likelier to have arisen from the dog's connection with birth than the dog's demonic associations.

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 learning about the construction of ancient eyes through fossil evidence is problematic due to the soft tissues leaving no imprint or remains, genetic and comparative anatomical evidence has increasingly supported the idea of a common ancestry for all eyes.

Although the crusaders came upon an ancient urban society, Ellenblum argues that they neither completely abandoned their rural European lifestyle, nor was European society completely rural to begin with.

Although the ancient designations are still adhered to, in modern Shinto many priests also consider kami to be anthropomorphic spirits, with nobility and authority.