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Although the axiom of countable choice in particular is commonly used in constructive mathematics, its use has also been questioned.
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Although and axiom
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 it was perhaps implied, he did not include an axiom requiring at least one subset to be independent.
Although the axiom of constructibility does resolve many set-theoretic questions, it is not typically accepted as an axiom for set theory in the same way as the ZFC axioms.
Although and countable
Although there are points in the Julia set whose sequence of iterations is finite, there are only a countable number of such points ( and they make up an infinitely small part of the Julia set ).
Although the usual base of open balls is not countable, one can restrict to the set of all open balls with rational radii and whose centers have rational coordinates.
Although and choice
Although he described his method as translating " sometimes word for word, sometimes sense for sense ," Alfred's translation actually keeps very close to his original, although through his choice of language he blurred throughout the distinction between spiritual and secular authority.
Although collectively this choice was made freely, religious Jews believe that it created individual obligation for the descendants of the Israelites.
Although killing oneself is forbidden under normal Jewish law as being a denial of God's goodness in the world, under extreme circumstances when there has seemed no choice but to either be killed or forced to betray their religion, Jews have committed suicide or mass suicide ( see Masada, First French persecution of the Jews, and York Castle for examples ).
Although the fundamental group in general depends on the choice of base point, it turns out that, up to isomorphism ( actually, even up to inner isomorphism ), this choice makes no difference as long as the space X is path-connected.
Although Major proved " a great disappointment to Thatcher ," he was her preferred choice as successor as she expected to " continue in control of the country as a backseat driver ".
Although not particular in the choice of some of the associates of his pleasures, he admitted none but men of worth to his intimacy, and when once admitted they were treated like equals.
Although he grew up in a middle class Jewish family, Altenberg eventually separated himself from his family of origin by dropping out of both law and medical school, and embracing Bohemianism as a permanent lifestyle choice.
Although parchment never stopped being used ( primarily for governmental documents and diplomas ) it had ceased to be a primary choice for artist's supports by the end of 15th century Renaissance.
Although models used in rational choice theory are diverse, all assume individuals choose the best action according to unchanging and stable preference functions and constraints facing them.
Although Worf took immense pride and a sense of honor from serving in Starfleet, most other Klingons shunned and belittled his choice of vocation.
Although he did not attend the 1912 Democratic National Convention in Baltimore, his name was put forward as Indiana's choice for president.
Although it was not a great success at the time – selling fewer than 3, 000 copies in the United States during 1955 before going out of print – it soon went on to become a best-seller, and by the early 1960s was required reading in many schools and colleges ; the novel is currently renowned for being a popular choice of study for GCSE English Literature courses in the United Kingdom.
Although sceptical of Albarn's new manifesto for the band, Balfe gave assent for the band's choice of Andy Partridge of the band XTC to produce their follow-up to Leisure.
Although Reagan would win a landslide victory, the polls showed the two major party candidates even ( the New York Times had it 44-43-8 ) going into the last weekend and it was clear that many would-be Anderson supporters were now supporting their second choice.
Although The Flowering Peach by Clifford Odets was the preferred choice of the Pulitzer Prize jury in 1955 and Cat on a Hot Tin Roof was at first considered the weakest of the five shortlisted nominees, Joseph Pulitzer Jr., chairman of the Board, had seen Cat on a Hot Tin Roof and thought it worthy of the drama prize.
Although initially regarded as a joinery wood, mahogany rapidly became the timber of choice for makers of high quality furniture in both the British Isles and the 13 colonies of North America.
Although shorter swords and daggers were used, the katana was probably the ninja's weapon of choice, and was sometimes carried on the back.
Although Kirk throughout the series becomes romantically involved with various women, when confronted with a choice between a woman and the Enterprise, " his ship always won ".
The importance of language in Swiss hip hop can also create tension, however: Although the members of the Italian speaking group Stoffunita live in Switzerland and consider it their home, their choice of language combined with their lack of Swiss citizenship earmark them as " Secondo ", a term used to indicate people of foreign descent born in Switzerland.
Although several Māori names have been used, Maori Language Commissioner Erima Henare sees Te Ika-a-Māui as the most likely choice.
Although several Māori names have been used, Maori Language Commissioner Erima Henare sees Te Wai Pounamu as the most likely choice.
Although Van Rensselaer had severe doubts about the quality of his troops, he had no choice but to attack.
Although and particular
Although they were forced to maintain a sharper watch, this activity enabled them to ride in and rack their broncs without any particular attention being paid them.
Although the particular form of conceptualization which popular imagination had made in response to the experience of spirit was undoubtedly defective, the raw experience itself which led to such excesses remains with us as vividly as ever.
Although " android " is used almost universally to refer to both sexes, and those of no particular sex, " android " technically refers to the male form, while " gynoid " is the feminine form.
Although the name actually applies to a group of snakes, it is often used to refer only to one species in particular, the common or green anaconda, Eunectes murinus, which is one of the largest snakes in the world.
Although often cited as a proponent of existentialism, the philosophy with which Camus was associated during his own lifetime, he rejected this particular label.
Although much of Inge's principles, above, still apply to the New Agrarianism, the affiliation with a particular religion and patriarchal tendency have subsided to some degree.
Although he is sometimes regarded as among the first and most influential architects of Nordic modernism, a closer examination of the historical facts reveals that Aalto ( while a pioneer in Finland ) closely followed and had personal contacts with other pioneers in Sweden, in particular Gunnar Asplund and Sven Markelius.
Although each variant of bridge has its own particular scheme for awarding and accumulating points, all are based upon whether or not the contract for each deal was made or defeated and by how many tricks.
Although there is no agreement on how a city is distinguished from a town within general English language meanings, many cities have a particular administrative, legal, or historical status based on local law.
Although the Church of the Holy Sepulchre has its tomb just a few yards away from its Golgotha, there is no particular reason to regard this close juxtaposition as a necessity ; however, Gordon followed this principle, concluding that his site for Golgotha must also be the approximate location for Jesus ' burial, identifying a nearby tomb, now called the Garden Tomb, as the location for the event.
Although several versions of many-worlds have been proposed since Hugh Everett's original work, they all contain one key idea: the equations of physics that model the time evolution of systems without embedded observers are sufficient for modelling systems which do contain observers ; in particular there is no observation-triggered wave function collapse which the Copenhagen interpretation proposes.
Although the details of each particular elliptic curve scheme are described in the article referenced above some common implementation considerations are discussed here.
Although the words faith and belief are sometimes erroneously conflated and used as synonyms, faith properly refers to a particular type ( or subset ) of belief, as defined above.
Although the exchange interaction keeps spins aligned, it does not align them in a particular direction.
Although, in the United Kingdom ( UK ), the pejorative noun " anorak " ( similar to the Japanese " otaku ", meaning a geek or enthusiast ) is often applied to people who obsessively pursue a particular hobby that is otherwise considered boring.
Although all the preparatory work for the decimal changeover had been done while Menzies was Prime Minister, Holt had particular responsibility as Treasurer for currency matters, and he was centrally involved in both the decision to change and its implementation.
Although the practice of Holy Communion was first alluded to in the Christian Bible and defined by theologians in the 1st centuries AD, it was around the time of the appearance of the first Christianised Grail literature that the Roman church was beginning to add more ceremony and mysticism around this particular sacrament.
Although they still give an overall score, they now also give scores for many of these more restricted abilities, identifying particular strengths and weaknesses of an individual.
Until the day I die people will keep saying, ‘ Leni is a Nazi ’, and I ’ ll keep saying, ‘ But what did she do ?’” Although she won more than 50 libel cases against people accusing her of collaborating with the Nazis, there are many unanswered questions about her relation to National Socialism in particular and fascism more generally.
Although the hare does not represent any particular figure in Greek mythology, Lepus was one of the 48 constellations listed by the 2nd century astronomer Ptolemy, and it remains one of the 88 modern constellations.
Although some books ( in particular guidebooks ) suggest that the mazes on cathedral floors served as substitutes for pilgrimage paths, the earliest attested use of the phrase " chemin de Jerusalem " ( path to Jerusalem ) dates to the late 18th century when it was used to describe mazes at Reims and Saint-Omer.
Although these agencies are ' global ' in the sense of their activities, they each retain significant associations with particular nations, namely France ( AFP ), the United States ( AP ) and the United Kingdom ( Reuters ).
Although power-law relations are attractive for many theoretical reasons, demonstrating that data do indeed follow a power-law relation requires more than simply fitting a particular model to the data.