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Hurrians

The Hurrians (also called Hari, Khurrites, Hourri, Churri, Hurri) were a people who inhabited the Ancient Near East during the Bronze Age. They spoke the Hurrian language, and lived throughout northern Syria, upper Mesopotamia and southeastern Anatolia.

The Hurrians were first documented in the city of Urkesh, where they built their first kingdom. Their largest and most influential Hurrian kingdom was Mitanni. The population of the Hittite Empire in Anatolia included a large population of Hurrians, and there is significant Hurrian influence in Hittite mythology. By the Early Iron Age, the Hurrians had been assimilated with other peoples. The state of Urartu later covered some of the same area.

Early Bronze Age
The Khabur River valley became the heart of the Hurrian lands for a millennium. The first known Hurrian kingdom emerged around the city of Urkesh (modern Tell Mozan) during the third millennium BC. There is evidence that they were initially allied with the Akkadian Empire of Mesopotamia, indicating they had a firm hold on the area by the reign of Naram-Sin of Akkad (c. 2254–2218 BC). A king of Urkesh with the Hurrian name Tupkish had a queen with the name Uqnitum, Akkadian for "girl of lapis lazuli".

Middle Bronze Age
Hurrian names occur sporadically in northwestern Mesopotamia and the area of Kirkuk in modern Iraq by the Middle Bronze Age. Their presence was attested at Nuzi, Urkesh and other sites. They eventually occupied a broad arc of fertile farmland stretching from the Khabur River valley in the west to the foothills of the Zagros Mountains in the east. By this point, during the Old Babylonian period in the early second millennium BC, the Amorite kingdom of Mari to the south had subdued Urkesh and made it a vassal state. Urkesh later became a Mitanni religious center.

The Hurrians also migrated further west in this period. By 1725 BC they are found also in parts of northern Syria, such as Alalakh. The mixed Amorite–Hurrian kingdom of Yamhad is recorded as struggling for this area with the early Hittite king Hattusilis I around 1600 BC. Hurrians also settled in the coastal region of Adaniya in the country of Kizzuwatna, southern Anatolia. Yamhad eventually weakened vis-a-vis the powerful Hittites, but this also opened Anatolia for Hurrian cultural influences. The Hittites were influenced by both the Hurrian cultures over the course of several centuries.

The city of Shibaniba (Tell Billa) may have also played an important role at that time. Possible Hurrian occupation was identified at Tell Billa during the middle of the second millenium BC. In 2022 Tell Billa was proposed as the possible site of the city of Šimānum (possibly known as Asimānum during the Akkadian Empire). Šimānum was important during the Ur III period (ca 2100 BC).

Late Bronze Age
The Mitanni Empire was a strong regional power limited by the Hittites to the north, Egyptians to the southwest, Kassites to the southeast, and later by the Assyrians to the east. At its maximum extent Mitanni ranged as far as west as Kizzuwatna by the Taurus mountains, Tunip in the south, Arraphe in the east, and north to Lake Van. Their sphere of influence is shown in spread Hurrian place names, personal names. Eventually, after an internal succession crisis, Mitanni fell to the Hittites, later to fall under the control of the Assyrians.

The Hurrian entity of Mitanni, which first rose to power before 1550 BC, was first mentioned in the records of Egyptian pharaohs Thutmose I (1506–1493 BC) and Thutmose III (1479–1425 BC), the later most notably associated with the Battle of Megiddo in that pharoahs 22 regnal year. Most of the time Egyptians referred to the kingdom as Naharin. Later, Mitanni and Hanigailbat are mention in the Amarna Letters during the time of Pharaoh Akhenaten (1353–1336 BC). Domestically, Mitanni records have been found at a number of places in the region including several Hittite sites as well as Tell Bazi, Alalakh, Nuzi, Mardaman, Kemune, and Müslümantepe among others.

Another major center of Hurrian influence was the kingdom of Arrapha. Excavations at Yorgan Tepe, ancient Nuzi, proved this to be one of the most important sites for our knowledge about the Hurrians. Hurrian kings such as Ithi-Teshup and Ithiya ruled over Arrapha, yet by the mid-fifteenth century BC they had become vassals of the Great King of Mitanni.

Urartu
At the end of the second millennium BC the Urartians around Lake Van and Mount Ararat rose in power forming the Kingdom of Urartu. During the 11th and 10th centuries BC, the kingdom eventually encompassed a region stretching from the Caucasus Mountains in the north, to the borders of northern Assyria and northern Ancient Iran in the south, and controlled much of eastern Anatolia. Some scientists consider Urartu to be a re-consolidation of earlier Hurrian populations mainly due to linguistic factors, but this view is not universally held.

Shubaru/Shubria
After the destruction of Mitanni by the Hittites around 1350-1325 BC, the term Shubaru was used in Assyrian sources to refer to the remnants of the Mitanni in the upper Tigris valley. The Shubaru people revolted against the Assyrians multiple times in the last centuries of the second millennium BC. The term is related to Shubria, the name of a country located north of the upper Tigris River valley. Shubria was located between Urartu and Assyria and existed as an independent kingdom until its conquest by Assyria in 673–672 BC. The Shubrians worshipped the Hurrian deity Teshub, and several Shubrian names have Hurrian origins. Hurrians formed part of the Shubrian population and may have been the predominant group. Some scholars have suggested that Shubria was the last remnant of Hurrian civilization, or even constituted the original homeland of the Hurrians. Karen Radner writes that Shubria "can certainly be described as [a] (linguistically and culturally) Hurrian" state. According to Radner, a letter from the king of Shubria to an Assyrian magnate from the time of Sargon II was composed in the Hurrian language.

Culture and society

Knowledge of Hurrian culture relies on archaeological excavations at sites such as Nuzi and Alalakh as well as on cuneiform tablets, primarily from Hattusa (Boghazköy), the capital of the Hittites, whose civilization was greatly influenced by the Hurrians. Tablets from Nuzi, Alalakh, and other cities with Hurrian populations (as shown by personal names) reveal Hurrian cultural features even though they were written in Akkadian. Hurrian cylinder seals were carefully carved and often portrayed mythological motifs. They are a key to the understanding of Hurrian culture and history.

The 2nd millennium Hurrians were masterful ceramists. Their pottery is commonly found in Mesopotamia and in the lands west of the Euphrates; it was highly valued in distant Egypt, by the time of the New Kingdom. Archaeologists use the terms Khabur ware and Nuzi ware for two types of wheel-made pottery used by the Hurrians. Khabur ware is characterized by reddish painted lines with a geometric triangular pattern and dots, while Nuzi ware has very distinctive forms, and are painted in brown or black. They were also skilled at glass working.

The Hurrians had a reputation in metallurgy. It is proposed that the Sumerian term for "coppersmith" tabira/tibira was borrowed from Hurrian, which would imply an early presence of the Hurrians way before their first historical mention in Akkadian sources. Copper was traded south to Mesopotamia from the highlands of Anatolia. The Khabur Valley had a central position in the metal trade, and copper, silver and even tin were accessible from the Hurrian-dominated countries Kizzuwatna and Ishuwa situated in the Anatolian highland. Gold was in short supply, and the Amarna letters inform us that it was acquired from Egypt. Not many examples of Hurrian metal work have survived, except from the later Urartu. Some small fine bronze lion foundation pegs were discovered at Urkesh.

Among the Hurrian texts from Ugarit are the oldest known instances of written music, dating from c. 1400 BC. Among these fragments are found the names of four Hurrian composers, Tapšiẖuni, Puẖiya(na), Urẖiya, and Ammiya.

Religion
The Hurrian culture made a great impact on the religion of the Hittites. From the Hurrian cult centre at Kummanni in Kizzuwatna, Hurrian religion spread to the Hittite people. Syncretism merged the Old Hittite and Hurrian religions. Hurrian religion spread to Syria, where Baal became the counterpart of Teshub. The Hurrian religion, in different forms, influenced the entire ancient Near East, except ancient Egypt and southern Mesopotamia.

While the Hurrian and Urartian languages are related, there is little similarity between corresponding systems of belief.

The main gods in the Hurrian pantheon were:
 * Teshub, Teshup, the mighty weather god.
 * Hebat, Hepa, his wife, the mother goddess, later equated with the main sun goddess of the Hittites
 * Sarruma, Šarruma, their son, a mountain god of Syrian origin.
 * Kumarbi, grain god, the father of Teshub and a "father of gods" similar to Enlil; his home as described in mythology is the city of Urkesh.
 * Shaushka, Šauska, the Hurrian counterpart of Ishtar, and a goddess of love, war and healing.
 * Shimegi, Šimegi, the sun god.
 * Kushuh, Kušuh, the moon god and a guardian of oaths. Symbols of the sun and the crescent moon appear joined together in the Hurrian iconography.
 * Nergal, a Sumerian deity of the netherworld, who had a prominent temple in Urkesh in the earliest period of recorded Hurrian history. Possibly a stand-in for a god whose Hurrian name is presently unknown.
 * Ea, Hayya, the god of wisdom, who was also Sumerian in origin.
 * Allani, goddess of the netherworld.
 * Ishara, a goddess of Syrian origin.
 * Aštabi, a war god.
 * Nupatik, a prominent god of uncertain function.
 * Hutena and Hutellura, fate and birth goddesses.

Hurrian cylinder seals often depict mythological creatures such as winged humans or animals, dragons and other monsters. The interpretation of these depictions of gods and demons remains uncertain. They may have been both protective and evil spirits. Some are reminiscent of the Assyrian shedu.

The Hurrian gods do not appear to have had particular home temples, like in the Mesopotamian or Ancient Egyptian religion. Some important cult centres were Kummanni in Kizzuwatna and Hittite Yazilikaya. Harran was at least later a religious centre for the moon god, and Shauskha had an important temple in Nineve, when the city was under Hurrian rule. A temple of Nergal was built in Urkesh in the late third millennium BC. The town of Kahat was a religious centre in the kingdom of Mitanni.

The Hurrian myth "The Songs of Ullikummi", preserved among the Hittites, is a parallel to Hesiod's Theogony; the castration of Uranus by Cronus may be derived from the castration of Anu by Kumarbi, while Zeus's overthrow of Cronus and Cronus's regurgitation of the swallowed gods is like the Hurrian myth of Teshub and Kumarbi. It has been argued that the worship of Attis drew on Hurrian myth.

Language

The agglutinating and highly ergative Hurrian language is related to the Urartian language, the language of the ancient kingdom of Urartu. Together they form the Hurro-Urartian language family. The external connections of the Hurro-Urartian languages are disputed. There exist various proposals for a genetic relationship to other language families (e.g. the Northeast Caucasian languages), but none of these are generally accepted.

The Hurrians adopted the Akkadian language and Cuneiform script for their own writing about 2000 BC. Texts in the Hurrian language in cuneiform have been found at Hattusa, Ugarit (Ras Shamra), as well as in one of the longest of the Amarna letters (EA 27), written by King Tushratta of Mitanni to Pharaoh Amenhotep III. It was the only long Hurrian text known until a multi-tablet collection of literature in Hurrian with a Hittite translation was discovered at Hattusa in 1983.

Archaeology
Hurrian settlements are distributed over three modern countries, Iraq, Syria and Turkey. The heart of the Hurrian world is bisected by the modern border between Syria and Turkey. Several sites are situated within the border zone, making access for excavations problematic. A threat to the ancient sites are the many dam projects in the Euphrates, Tigris and Khabur valleys. Several rescue operations have already been undertaken when the construction of dams put entire river valleys under water.

The first major excavations of Hurrian sites in Iraq and Syria began in the 1920s and 1930s. They were led by the American archaeologist Edward Chiera at Yorghan Tepe (Nuzi), and the British archaeologist Max Mallowan at Chagar Bazar and Tell Brak. Recent excavations and surveys in progress are conducted by American, Belgian, Danish, Dutch, French, German and Italian teams of archaeologists, with international participants, in cooperation with the Syrian Department of Antiquities. The tells, or city mounds, often reveal a long occupation beginning in the Neolithic and ending in the Roman period or later. The characteristic Hurrian pottery, the Khabur ware, is helpful in determining the different strata of occupation within the mounds. The Hurrian settlements are usually identified from the Middle Bronze Age to the end of the Late Bronze Age, with Tell Mozan (Urkesh) being the main exception.

Important sites
The list includes some important ancient sites from the area dominated by the Hurrians. Excavation reports and images are found at the websites linked. As noted above, important discoveries of Hurrian culture and history were also made at Alalakh, Amarna, Hattusa and Ugarit.

• Urkesh (Tell Mozan)

• Nuzi (Yorghan Tepe)

• Nagar (Tell Brak)

• Shehna and Shubat-Enlil (Tell Leilan)

• Kahat (Tell Barri)

• Nabada (Tell Beydar)

• Tell Tuneinir

• Umm el-Marra (Tuba?)

• Tell Chuera

• Tell Hammam et-Turkman (Zalpa?)

• Tell Sabi Abyad

• Hamoukar

• Chagar Bazar

• Tell Fekheriye (Sikkan, Washukanni?)

• Tall Al-Hamidiya (Taidu?)

title: Hurrians


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Cook (surname)

Cook is an occupational surname of English origin. Notable people with the surname include:

A

 * Aaron Cook (disambiguation), multiple people
 * Adam Cook (disambiguation), multiple people
 * A. J. Cook (born 1978), Canadian actress
 * A. J. Cook (trade unionist) (1883–1931), Welsh trade unionist
 * Alan Cook (disambiguation) or Allan, multiple people
 * Alana Cook (born 1997), American soccer player
 * Alastair Cook (born 1984), English cricketer
 * Albert Cook (disambiguation), multiple people
 * Alex Cook (disambiguation), multiple people
 * Alfred M. Cook, American politician
 * Ali Cook, English magician and actor
 * Alice Cook (disambiguation), multiple people
 * Alicia Augello Cook also known as Alicia Keys, American singer
 * Alistair Cook (disambiguation), multiple people
 * Allison Cook (disambiguation), multiple people
 * Alyssa-Jane Cook (born 1967), Australian actress, singer and television presenter
 * Amanda Lindsey Cook (born 1984), Canadian singer
 * Amy Cook (born 1979), American musician and singer-songwriter
 * Anne Cook (disambiguation), or Ann, multiple people
 * Annelies Cook (born 1984), American biathlete
 * Anthony Cook (disambiguation), multiple people
 * Arnold Cook (1922–1981), Australian academic and national guide dog pioneer
 * Arthur Cook (disambiguation) multiple people
 * Austin Cook (born 1991), American golfer

B

 * Barbara Cook (1927–2017), American singer and actress
 * Barrie Cook (1929–2020), English artist
 * Barry Cook, American film director
 * Beano Cook (1931–2012), American sportscaster
 * Bekki Cook (born 1950), American politician
 * Benjamin Cook (disambiguation), multiple people
 * Bert Cook (disambiguation), multiple people
 * Beryl Cook (1926–2008), English painter
 * Betsy Cook, American musician
 * Bill Cook or Billy Cook see William Cook (disambiguation)
 * Blaine Cook, American singer
 * Blaine Cook (programmer), Canadian software engineer
 * Bob Cook (1948–1978), Canadian ice hockey player
 * Bobby Cook (disambiguation), several people
 * Boo Cook (born 1972), British comic artist
 * Brendon Cook (born 1979), Australian racing driver
 * Brian Cook (disambiguation), multiple people
 * Bruce Cook (disambiguation), several people
 * Bun Cook, Canadian professional ice hockey forward
 * Byron Cook (disambiguation), several people

C

 * Carole Cook (1924–2023), American actress
 * Carl Cook (born 1962), American businessman
 * Catherine Cook (disambiguation), multiple people
 * Charles Cook (disambiguation), multiple people
 * Chris Cook (disambiguation), multiple people
 * Christine Cook (born 1970), English field hockey player
 * Clara Cook (1921–1996), American baseball player
 * Clay Cook (born 1978), American songwriter, producer and musician

D

 * Daequan Cook (born 1987), American player with Israeli Basketball Premier League
 * Dale Cook (born 1958), American kickboxer
 * Dalvin Cook (born 1995), American football player
 * Daniel Cook (disambiguation), multiple people
 * Damien Cook (born 1991), Australian Rugby League player
 * Dane Cook (born 1972), American comedian
 * David Cook (disambiguation), multiple people
 * Dean Cook (born 1985), British actor
 * Dennis Cook (born 1962), American professional baseball pitcher
 * Dianne Cook (disambiguation), multiple people
 * Dick Cook, American Chairman of Disney Studios
 * Doc Cook (Charles L. Cooke, 1891–1958), American jazz bandleader
 * Doc Cook (baseball) (Luther Almus Cook, 1886–1973), American baseball player
 * Don Cook (born 1949), American record producer and songwriter, mainly in the field of country music
 * Donald Cook (disambiguation), multiple people
 * Donna Cook (1926–2006), All-American Girls Professional Baseball League player
 * Doris Cook (born 1931), All-American Girls Professional Baseball League player
 * Dorothy Cook, All-American Girls Professional Baseball League player
 * Dylan Cook (born 1998), American football player

E

 * Edith Agnes Cook (1859–1942), first female student at Adelaide University, principal of the Advanced School for Girls
 * Eli Cook (1814–1865), American politician
 * Eli Cook (musician) (born 1986), American singer, songwriter, guitarist and record producer
 * Elisha Cook Jr. (1903–1995), American actor
 * Elizabeth Cook (disambiguation), multiple people
 * Everett Richard Cook (1894–1974), American World War I flying ace

F

 * Fielder Cook (1923–2003), American television and film director, producer and writer
 * Francis Cook (disambiguation), multiple people
 * Frank Cook (disambiguation), multiple people
 * Fred Cook (disambiguation), multiple people
 * Frederick Cook (disambiguation), multiple people

G

 * Garry Cook (born 1958), British athlete
 * Garry Cook (CEO), British football executive, currently with Manchester City
 * George Cook (disambiguation), multiple people
 * Geoff Cook (born 1951), England cricketer
 * Geoffrey Cook (disambiguation), multiple people
 * Glen Cook (born 1944), author
 * Glen Cook (baseball), American baseball player
 * Glenn Cook, British triathlete and coach
 * Greg Cook (disambiguation), multiple people

H

 * Harry Cook (disambiguation), multiple people
 * Henry Cook (disambiguation), multiple people
 * Herman Cook, known as Junior Cook (1934–1992), hard bop tenor saxophone player
 * Hugh Cook (Canadian novelist) (born 1942), Canadian writer
 * Hugh Cook (science fiction author) (1956–2008), English-born novelist
 * Hume Cook, James Hume Cook, Australian politician

I

 * Ian Cook (disambiguation), multiple people
 * Ira Cook (1821–1902), American surveyor, mayor, banker, tax collector, city council member, investor and entrepreneur

J

 * James Cook (disambiguation), multiple people
 * Jamie Cook, guitarist with British band the Arctic Monkeys
 * Jamie Cook (footballer), English footballer
 * Jane Constance Cook (Ga’axstal’as) (1870–1951), Canadian first nations leader and activist
 * Jesse Cook, Canadian guitarist
 * Jeffrey Cook (disambiguation), multiple people
 * Jill Cook, British curator
 * Jim Cook see James Cook (disambiguation)
 * Jimmy Cook, South African cricketer
 * Joan Cook (born 1934), Canadian Senator for Newfoundland and Labrador
 * Joan Riddell Cook, founding director of JAWS (Journalism and Women Symposium)
 * Joel Cook (1842–1910), American politician from Pennsylvania
 * John Cook (disambiguation), multiple people
 * Johnny Cook (disambiguation), multiple people
 * Jonathan Cook (born 1965), British writer and journalist based in Israel
 * Jonathan Cook (cricketer) (born 1989), Australian cricketer
 * Jordan Cook (born 1990), English footballer
 * Joseph Cook (disambiguation), multiple people
 * Julian Cook (1916–2017), US Army officer
 * Julian Abele Cook Jr. (1930–2017), US District Court judge
 * Junior Cook (1934–1992), Herman Cook, American hard bop tenor saxophone player
 * Justin Cook (born 1982), American voice actor

K

 * Kathy Smallwood-Cook (born 1960), British sprint athlete
 * Kenneth Bernard Cook (1929–1987), Australian journalist and novelist
 * Kristy Lee Cook (born 1984), American singer

L

 * Lawrence Cook (disambiguation), multiple people
 * Lee Cook (born 1982), English footballer
 * Lemuel Cook (1759–1866), American Revolutionary War veteran
 * Leonard Cook (1912–?), Australian boxer at the 1936 Olympics
 * Lester Cook (born 1984), American tennis player
 * Lewis Cook (disambiguation), multiple people
 * Linda Cook (1962–1986), English murder victim
 * Louise Cook (born 1987), British rally driver

M

 * Mariana Cook (born 1955), American photographer
 * Marlow Cook (1926–2016), Republican US Senator from Kentucky
 * Marv Cook (born 1966), American football player
 * Mary Cook (disambiguation), multiple people
 * Matthew Cook (disambiguation), multiple people
 * Merrill Cook (born 1946), Republican member of the US House of Representatives from Utah
 * Michael Cook (disambiguation), multiple people
 * Myke Cook (born 1989), Australian football player
 * Mindy Cook (born 1988), American goalball player
 * Mitch Cook (born 1961), English football player
 * Moxon Cook (1857–1917), Australian sports journalist
 * Murray Cook (born 1960), Australian musician
 * Murray Cook (baseball) (born 1940), Canadian baseball scout and executive

N

 * Nat Cook, Australian politician
 * Natalie Cook (born 1975), Australian beach volleyball player
 * Nathaniel Cook (19th century), designer of the standard set of chess figures
 * Nicholas Cook (born 1950), British musicologist
 * Nick Cook (disambiguation), multiple people
 * Nicky Cook, English boxer
 * Norm Cook, American basketball player
 * Norman Cook (born 1963), British musician (aka Fatboy Slim)

O

 * Octavia Cook (born 1978), New Zealand jeweller
 * Olive Cook (1912–2002), British writer and artist
 * Oliver Cook (born 1990), British rower
 * Omar Cook (born 1982), American-born, naturalized Montenegrin professional basketball player
 * Omar Cook (American football) (born 1992), American football defensive back
 * Orator F. Cook (1867–1949), American botanist, entomologist, and agronomist
 * Orchard Cook (1763–1819), U.S. Representative from Massachusetts
 * Orval R. Cook (1898–1980), United States Air Force four-star general
 * Oscar Cook (1888–1952), British author of novels, non-fiction works and short stories with a supernatural theme
 * Óscar Alvarado Cook (born 1944), Mexican politician from the Institutional Revolutionary Party
 * Otis Cook (1900–1980), American painter born in New Bedford, Massachusetts

P

 * Pam Cook (born 1943), English film theorist
 * Pamela Cook, American mathematician
 * Paul Cook (disambiguation) multiple people
 * Paula Cook (born 1969), British auto racing driver
 * Penny Cook (1957–2018), Australian actress
 * Perry R. Cook (born 1955), American computer music researcher and professor at Princeton
 * Peter Cook (disambiguation), multiple people

Q

 * Quentin L. Cook (born 1940), American lawyer, business executive, and religious leader in the LDS Church
 * Quinn Cook (born 1993), American basketball player

R

 * R. James Cook (born 1937), American phytopathologist
 * Rachael Leigh Cook (born 1979), American actress and model
 * Ralph Cook (born 1944), Associate Justice of the Alabama Supreme Court
 * Randall William Cook (born 1951), American visual effects artist
 * Richard Cook (disambiguation) multiple people
 * Rick Cook (architect) (born 1960), New York City architect
 * Rick Cook (writer) (1944–2022), American author
 * Robin Cook (disambiguation), multiple people
 * Roderick Cook (1932–1990), English playwright, writer, director, and actor
 * Rodney Mims Cook Jr. (born 1956), American businessman
 * Rodney Mims Cook Sr. (1924–2013), American politician and activist
 * Roger Cook (disambiguation), multiple people
 * Ron Cook (born 1948), English actor
 * Ron Cook (baseball) (born 1947), American baseball player
 * Russ Cook, English endurance athlete
 * Ryan Cook (disambiguation), multiple people

S

 * Samuel Cook (disambiguation), multiple people
 * Scott Cook (born 1952), American entrepreneur
 * Scott Cook (soccer), American soccer player
 * Sharon Anne Cook (born 1947), Canadian historian
 * Sherburne F. Cook (1896–1974), American physiologist and pioneer in population studies
 * Stephen Cook (disambiguation)
 * Steve Cook (disambiguation), multiple people
 * Stu Cook (born 1945), bassist with U.S rock band Creedence Clearwater Revival
 * Stuart W. Cook (1913–1993), American social psychologist

T

 * Ted Cook (disambiguation), multiple people
 * Terry Cook (disambiguation), multiple people
 * Theodore Cook (disambiguation), multiple people
 * Thomas Cook (disambiguation), multiple people
 * T. S. Cook (1947–2013), American screenwriter and producer
 * Tim Cook (disambiguation), multiple people
 * Troy Cook (born 1976), Australian footballer
 * Tyler Cook (born 1997), American basketball player

V

 * Vernon Cook (1927–1987), former member of the Ohio House of Representatives
 * Victor Cook (born c. 1960/1961), American television director, writer, and producer
 * Victoria Cook (1933–2019), former World Champion archer who represented the United States
 * Virgil Young Cook (1848–1922), American Confederate veteran and planter
 * Vivian Cook (disambiguation), multiple people

W

 * Walter Cook (disambiguation), multiple people
 * Wayne Cook (American football) (born 1971), American football player
 * Wayne Cook (musician) (born 1946), American keyboardist
 * Wesley Cook (born 1954), later known as Mumia Abu-Jamal, American convicted of the 1981 murder of a Philadelphia police officer
 * Whitfield Cook (1909–2003), American writer
 * Will Marion Cook (1869–1944), American composer and violinist
 * William Cook (disambiguation), multiple people
 * Willie Cook (1923–2000), American jazz trumpeter
 * Willie Cook (footballer) (1903–1981), Scottish footballer
 * W. T. Cook (1884–1970), American college sports coach

Y

 * Yassar Cook (born 1993), South African first-class cricketer
 * Yvette Cook, known as "Lady E", member of Newcleus, American electro and old school hip hop group

Z

 * Zachary Cook (fl. 2000s–2010s), American politician, Republican member of the New Mexico House of Representatives
 * Zadock Cook (1769–1863), United States Representative from Georgia

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: Cook (surname)


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Thornton Wilder

Thornton Niven Wilder (April 17, 1897 – December 7, 1975) was an American playwright and novelist. He won three Pulitzer Prizes, for the novel The Bridge of San Luis Rey and for the plays Our Town and The Skin of Our Teeth, and a U.S. National Book Award for the novel The Eighth Day.

Early life and education
Wilder was born in Madison, Wisconsin, the son of Amos Parker Wilder, a newspaper editor and later a U.S. diplomat, and Isabella Thornton Niven.

Wilder had four siblings as well as a twin who was stillborn. All of the surviving Wilder children spent part of their childhood in China when their father was stationed in Hong Kong and Shanghai as U.S. Consul General. Thornton's older brother, Amos Niven Wilder, became Hollis Professor of Divinity at the Harvard Divinity School. He was a noted poet and was instrumental in developing the field of theopoetics. Their sister Isabel Wilder was an accomplished writer. They had two more sisters, Charlotte Wilder, a poet, and Janet Wilder Dakin, a zoologist.

Education
Wilder began writing plays while at the Thacher School in Ojai, California, where he did not fit in and was teased by classmates as overly intellectual. According to a classmate, "We left him alone, just left him alone. And he would retire at the library, his hideaway, learning to distance himself from humiliation and indifference." His family lived for a time in China, where his sister Janet was born in 1910. He attended the English China Inland Mission Chefoo School at Yantai, but returned with his mother and siblings to California in 1912 because of the unstable political conditions in China at the time. Thornton graduated from Berkeley High School in 1915.

Wilder served a three-month enlistment in the U.S. Army's Coast Artillery Corps at Fort Adams in Rhode Island during World War I, eventually rising to the rank of corporal. He attended Oberlin College before earning his Bachelor of Arts degree in 1920 at Yale University, where he refined his writing skills as a member of the Alpha Delta Phi fraternity, a literary society. He earned his Master of Arts degree in French literature from Princeton University in 1926.

Career
After graduating, Wilder went to Italy and studied archaeology and Italian (1920–21) as part of an eight-month residency at The American Academy in Rome, and then taught French at the Lawrenceville School in Lawrenceville, New Jersey, beginning in 1921. His first novel, The Cabala, was published in 1926. In 1927, The Bridge of San Luis Rey brought him commercial success and his first Pulitzer Prize (1928). He resigned from the Lawrenceville School in 1928. From 1930 to 1937 he taught at the University of Chicago, during which time he published his translation of André Obey's own adaptation of the tale "Le Viol de Lucrece" (1931) under the title "Lucrece" (Longmans Green, 1933). In Chicago, he became famous as a lecturer and was chronicled on the celebrity pages. In 1938 he won the Pulitzer Prize for Drama for his play Our Town, and he won the prize again in 1943 for his play The Skin of Our Teeth.

World War II saw Wilder rise to the rank of lieutenant colonel in the U.S. Army Air Force Intelligence, first in Africa, then in Italy until 1945. He received several awards for his military service. He went on to be a visiting professor at Harvard University, where he served for a year as the Charles Eliot Norton professor. Though he considered himself a teacher first and a writer second, he continued to write all his life, receiving the Peace Prize of the German Book Trade in 1957 and the Presidential Medal of Freedom in 1963. In 1968 he won the National Book Award for his novel The Eighth Day.

Proficient in four languages, Wilder translated plays by André Obey and Jean-Paul Sartre. He wrote the libretti of two operas, The Long Christmas Dinner, composed by Paul Hindemith, and The Alcestiad, composed by Louise Talma and based on his own play. Alfred Hitchcock, whom he admired, asked him to write the screenplay of his thriller Shadow of a Doubt, and he completed a first draft for the film.

The Bridge of San Luis Rey (1927) tells the story of several unrelated people who happen to be on a bridge in Peru when it collapses, killing them. Philosophically, the book explores the question of why unfortunate events occur to people who seem "innocent" or "undeserving". It won the Pulitzer Prize in 1928, and in 1998 it was selected by the editorial board of the American Modern Library as one of the 100 best novels of the twentieth century. The book was quoted by British Prime Minister Tony Blair during the memorial service for victims of the September 11 attacks in 2001. Since then its popularity has grown enormously. The book is the progenitor of the modern disaster epic in literature and film-making, where a single disaster intertwines the victims, whose lives are then explored by means of flashbacks to events before the disaster.

Wilder wrote Our Town, a popular play (and later film) set in fictional Grover's Corners, New Hampshire. It was inspired in part by Dante's Purgatorio and in part by his friend Gertrude Stein's novel The Making of Americans. Wilder suffered from writer's block while writing the final act. Our Town employs a choric narrator called the Stage Manager and a minimalist set to underscore the human experience. Wilder himself played the Stage Manager on Broadway for two weeks and later in summer stock productions. Following the daily lives of the Gibbs and Webb families, as well as the other inhabitants of Grover's Corners, the play illustrates the importance of the universality of the simple, yet meaningful lives of all people in the world in order to demonstrate the value of appreciating life. The play won the 1938 Pulitzer Prize.

In 1938, Max Reinhardt directed a Broadway production of The Merchant of Yonkers, which Wilder had adapted from Austrian playwright Johann Nestroy's Einen Jux will er sich machen (1842). It was a failure, closing after 39 performances.

His play The Skin of Our Teeth opened in New York on November 18, 1942, featuring Fredric March and Tallulah Bankhead. Again, the themes are familiar – the timeless human condition; history as progressive, cyclical, or entropic; literature, philosophy, and religion as the touchstones of civilization. Three acts dramatize the travails of the Antrobus family, allegorizing the alternate history of mankind. It was claimed by Joseph Campbell and Henry Morton Robinson, authors of A Skeleton Key to Finnegans Wake, that much of the play was the result of unacknowledged borrowing from James Joyce's last work.

In his novel The Ides of March (1948), Wilder reconstructed the characters and events leading to, and culminating in, the assassination of Julius Caesar. He had met Jean-Paul Sartre on a U.S. lecture tour after the war, and was under the influence of existentialism, although rejecting its atheist implications.

In 1954, Tyrone Guthrie encouraged Wilder to rework The Merchant of Yonkers into The Matchmaker. This time the play opened in 1955 and enjoyed a healthy Broadway run of 486 performances with Ruth Gordon in the title role, winning a Tony Award for Guthrie, its director. It became the basis for the hit 1964 musical Hello, Dolly!, with a book by Michael Stewart and score by Jerry Herman.

In 1960, Wilder was awarded the first ever Edward MacDowell Medal by The MacDowell Colony for outstanding contributions to American culture.

In 1962 and 1963, Wilder lived for 20 months in the small town of Douglas, Arizona, apart from family and friends. There he started his longest novel, The Eighth Day, which went on to win the National Book Award. According to Harold Augenbraum in 2009, it "attack[ed] the big questions head on, ... [embedded] in the story of small-town America".

His last novel, Theophilus North, was published in 1973, and made into the film Mr. North in 1988.

The Library of America republished all of Wilder's plays in 2007, together with some of his writings on the theater and the screenplay of Shadow of a Doubt. In 2009, a second volume was released, containing his first five novels, six early stories, and four essays on fiction. Finally, the third and final volume in the Library of America series on Wilder was released in 2011, containing his last two novels The Eighth Day and Theophilus North, as well as four autobiographical sketches.

Personal life
Six years after Wilder’s death, Samuel Steward wrote in his autobiography that he had sexual relations with him. In 1937, Gertrude Stein had given Steward, then a college professor, a letter of introduction to Wilder. According to Steward, Alice B. Toklas told him that Wilder liked him and that Wilder had reported he was having trouble starting the third act of Our Town until he and Steward walked around Zürich all night in the rain and the next day wrote the whole act, opening with a crowd in a rainy cemetery. Penelope Niven disputes Steward's claim of a relationship with Wilder and, based on Wilder's correspondence, says Wilder worked on the third act of Our Town over the course of several months and completed it several months before he first met Steward. Robert Gottlieb, reviewing Penelope Niven's work in The New Yorker in 2013, claimed Wilder had become infatuated with a man, not identified by Gottlieb, and Wilder’s feelings were not reciprocated. Gottlieb asserted that "Niven ties herself in knots in her discussion of Wilder’s confusing sexuality" and that "His interest in women was unshakably nonsexual." He takes Steward's view that Wilder was a latent homosexual but never comfortable with sex.

Wilder had a wide circle of friends, including writers Ernest Hemingway, F. Scott Fitzgerald, Zelda Fitzgerald, Toklas, Jean-Paul Sartre, and Stein; actress Ruth Gordon; fighter Gene Tunney; and socialite Sibyl, Lady Colefax.

From the earnings of The Bridge of San Luis Rey, in 1930 Wilder had a house built for his family in Hamden, Connecticut, designed by Alice Trythall Washburn, one of few female architects working at the time. His sister Isabel lived there for the rest of her life. This became his home base, although he traveled extensively and lived away for significant periods.

Death
Wilder died of heart failure in his Hamden, Connecticut house on December 7, 1975, at age 78. He is interred at Mount Carmel Cemetery in Hamden.

Plays

 * The Trumpet Shall Sound (1926)
 * The Angel That Troubled the Waters and Other Plays (1928):
 * "Nascuntur Poetae"
 * "Proserpina and the Devil"
 * "Fanny Otcott"
 * "Brother Fire"
 * "The Penny That Beauty Spent"
 * "The Angel on the Ship"
 * "The Message and Jehanne"
 * "Childe Roland to the Dark Tower Came"
 * "Centaurs"
 * "Leviathan"
 * "And the Sea Shall Give Up Its Dead"
 * "The Servant's Name Was Malchus"
 * "Mozart and the Gray Steward"
 * "Hast Thou Considered My Servant Job?"
 * "The Flight Into Egypt"
 * "The Angel That Troubled the Waters"
 * The Long Christmas Dinner and Other Plays in One Act (1931):
 * The Long Christmas Dinner
 * Queens of France
 * Pullman Car Hiawatha
 * Love and How to Cure It
 * Such Things Only Happen in Books
 * The Happy Journey to Trenton and Camden
 * Our Town (1938)—won the Pulitzer Prize for Drama
 * The Merchant of Yonkers (1938)
 * The Skin of Our Teeth (1942)—won the Pulitzer Prize
 * The Matchmaker (1954)—revised from The Merchant of Yonkers
 * The Alcestiad: Or, a Life in the Sun (1955)
 * Childhood (1960)
 * Infancy (1960)
 * Plays for Bleecker Street (1962)
 * The Collected Short Plays of Thornton Wilder Volume I (1997):
 * The Long Christmas Dinner
 * Queens of France
 * Pullman Car Hiawatha
 * Love and How to Cure It
 * Such Things Only Happen in Books
 * The Happy Journey to Trenton and Camden
 * The Drunken Sisters
 * Bernice
 * The Wreck on the Five-Twenty-Five
 * A Ringing of Doorbells
 * In Shakespeare and the Bible
 * Someone from Assisi
 * Cement Hands
 * Infancy
 * Childhood
 * Youth
 * The Rivers Under the Earth
 * Our Town

Films

 * Shadow of a Doubt (1943)

Novels

 * The Cabala (1926)
 * The Bridge of San Luis Rey (1927)—won the Pulitzer Prize for the Novel
 * The Woman of Andros (1930)—based on Andria, a comedy by Terence
 * Heaven's My Destination (1935)
 * Ides of March (1948)
 * The Eighth Day (1967)—won the National Book Award for Fiction
 * Theophilus North (1973)—reprinted as Mr. North following the appearance of the film of the same name

title: Thornton Wilder


text:
Auxiliary verb

An auxiliary verb (abbreviated ) is a verb that adds functional or grammatical meaning to the clause in which it occurs, so as to express tense, aspect, modality, voice, emphasis, etc. Auxiliary verbs usually accompany an infinitive verb or a participle, which respectively provide the main semantic content of the clause. An example is the verb have in the sentence I have finished my lunch. Here, the auxiliary have helps to express the perfect aspect along with the participle, finished. Some sentences contain a chain of two or more auxiliary verbs. Auxiliary verbs are also called helping verbs, helper verbs, or (verbal) auxiliaries. Research has been conducted into split inflection in auxiliary verbs.

Basic examples
Below are some sentences that contain representative auxiliary verbs from English, Spanish, German and French, with the auxiliary verb marked in bold:

 * a. Do you want tea? – do is an auxiliary accompanying the infinitive, want, used here to form a question – see do-support.

 * b. She has given her best shot. – have, from which has is inflected, is an auxiliary used in expressing the perfect aspect of give.

 * c. He cogido tu lápiz. – he is an auxiliary accompanying the infinitive coger, used here to form a verb phrase, the perfect present in Spanish.
 * (I) have grabbed your pencil = 'I have taken your pencil.'

 * d. Das wurde mehrmals gesagt. – werden, from which wurde is inflected, become is an auxiliary used to build the passive voice in German.
 * That became many times said = 'That was said many times.'

 * e. Sie ist nach Hause gegangen. – sein, from which ist is inflected, 'be' is an auxiliary used with movement verbs to build the perfect tense/aspect in German.
 * She is to home gone = 'She went home/She has gone home.'

 * f. J'ai vu le soleil. – avoir, from which ai is inflected, 'have' is an auxiliary used to build the perfect tense/aspect in French.
 * I have seen the sun = 'I have seen the sun/I saw the sun.'

 * g. Nous sommes hébergés par un ami. – être, from which sommes is inflected, 'be' is an auxiliary used to build the passive voice in French.
 * We are hosted by a friend.

These auxiliaries help express a question, show tense/aspect, or form passive voice. Auxiliaries like these typically appear with a full verb that carries the main semantic content of the clause.

Traits across languages
Auxiliary verbs typically help express grammatical tense, aspect, mood, and voice. They generally appear together with an infinitive. The auxiliary is said to "help" the infinitive. The auxiliary verbs of a language form a closed class, i.e., there is a fixed, relatively small number of them.

Widely acknowledged verbs that can serve as auxiliaries in English and many related languages are the equivalents of be to express passive voice, and have (and sometimes be) to express perfect aspect or past time reference.

In some treatments, the copula be is classed as an auxiliary even though it does not "help" another verb, e.g.,

 * The bird is in the tree. – is serves as a copula with a predicative expression not containing any other verb.

Definitions of auxiliary verbs are not always consistent across languages, or even among authors discussing the same language. Modal verbs may or may not be classified as auxiliaries, depending on the language. In the case of English, verbs are often identified as auxiliaries based on their grammatical behavior, as described below. In some cases, verbs that function similarly to auxiliaries, but are not considered full members of that class (perhaps because they carry some independent lexical information), are called semi-auxiliaries. In French, for example, verbs such as devoir (have to), pouvoir (be able to), aller (be going to), vouloir (want), faire (make), and laisser (let), when used together with the infinitive of another verb, can be called semi-auxiliaries. There has also been a study on auxiliary verb constructions in Dravidian languages.

English
The following sections consider auxiliary verbs in English. They list auxiliary verbs, then present the diagnostics that motivate this special class (subject-auxiliary inversion and negation with not). The modal verbs are included in this class, due to their behavior with respect to these diagnostics.

List of auxiliaries in English
A list of verbs that (can) function as auxiliaries in English is as follows:

 * be, can, could, dare, do, have, may, might, must, need, ought, shall, should, will, would

The status of dare (not), need (not), and ought (to) is debatable and the use of these verbs as auxiliaries can vary across dialects of English. If the negative forms can't, don't, won't, etc. are viewed as separate verbs (and not as contractions), then the number of auxiliaries increases. The verbs do and have can also function as full verbs or as light verbs, which can be a source of confusion about their status. The modal verbs (can, could, may, might, must, shall, should, will, would, and dare, need and ought when included) form a subclass of auxiliary verbs. Modal verbs are defective insofar as they cannot be inflected, nor do they appear as gerunds, infinitives, or participles.

The following table summarizes the auxiliary verbs in standard English and the meaning contribution to the clauses in which they appear. Many auxiliary verbs are listed more than once in the table based upon discernible differences in use.

Deontic modality expresses an ability, necessity, or obligation that is associated with an agent subject. Epistemic modality expresses the speaker's assessment of reality or likelihood of reality. Distinguishing between the two types of modality can be difficult, since many sentences contain a modal verb that allows both interpretations.

List of auxiliaries unique to African American Vernacular English
African American Vernacular English makes a variety of finer tense/aspect distinctions than other dialects of English by making use of unique variant forms of, in particular: habitual 'be', reduced 'done' (dən), and stressed 'been' (BIN):

Diagnostics for identifying auxiliary verbs in English
The verbs listed in the previous section can be classified as auxiliaries based upon two diagnostics: they allow subject–auxiliary inversion (the type of inversion used to form questions etc.) and (equivalently) they can take not as a postdependent (a dependent that follows its head). The following examples illustrate the extent to which subject–auxiliary inversion can occur with an auxiliary verb but not with a full verb. (The asterisk * is the means commonly used in linguistics to indicate that the example is grammatically unacceptable or that a particular construction has never been attested in use).

 * a. He was working today.
 * b. Was he working today? - Auxiliary verb was allows subject–auxiliary inversion.

 * a. He worked today.
 * b. *Worked he today? - Full verb worked does not allow subject–auxiliary inversion.

 * a. She can see it.
 * b. Can she see it? - Auxiliary verb can allows subject–auxiliary inversion.

 * a. She sees it.
 * b. *Sees she it? - Full verb sees does not allow subject–auxiliary inversion.

The following examples illustrate that the negation not can appear as a postdependent of a finite auxiliary verb, but not as a postdependent of a finite full verb:

 * a. Sam would try that.
 * b. Sam would not try that. - The negation not appears as a postdependent of the finite auxiliary would.

 * a. Sam tried that.
 * b. *Sam tried not that. - The negation not cannot appear as a postdependent of the finite full verb tried.
 * a. Tom could help.
 * b. Tom could not help. - The negation not appears as a postdependent of the finite auxiliary could.

 * a. Tom helped.
 * b. *Tom helped not. - The negation not cannot appear as a postdependent of the finite full verb helped.

A third diagnostic that can be used for identifying auxiliary verbs is verb phrase ellipsis. See the article on verb phrase ellipsis for examples.

These criteria lead to the copula be and non-copular use of be as an existential verb being considered an auxiliary (it undergoes inversion and takes postdependent not, e.g., Is she the boss?, She is not the boss, Is there a God?, There is a God). However, if one defines auxiliary verb as a verb that somehow "helps" another verb, then the copula be is not an auxiliary, because it appears without another verb. The literature on auxiliary verbs is somewhat inconsistent in this area.

There are also some properties that some but not all auxiliary verbs have. Their presence can be used to conclude that the verb is an auxiliary, but their absence does not guarantee the converse. One such property is to have the same form in the present tense, also for the first and the third person singular. This in particular is typical for modal auxiliary verbs, such as will and must. (Examples: He will come tomorrow, she must do it at once, not he wills or she musts .)

Light verbs
Some syntacticians distinguish between auxiliary verbs and light verbs. The two are similar insofar as both verb types contribute mainly just functional information to the clauses in which they appear. Hence both do not qualify as separate predicates, but rather they form part of a predicate with another expression – usually with a full verb in the case of auxiliary verbs and usually with a noun in the case of light verbs.

In English, light verbs differ from auxiliary verbs in that they cannot undergo inversion and they cannot take not as a postdependent. The verbs have and do can function as auxiliary verbs or as light verbs (or as full verbs). When they are light verbs, they fail the inversion and negation diagnostics for auxiliaries, e.g.

Note that in some dialects (for example, the West and South West dialects of Hiberno-English), the inversion test may sound correct to native speakers.
 * a. They had a long meeting.

 * b. *Had they a long meeting? - Light verb had fails the inversion test.

 * c. *They had not a long meeting. - Light verb had fails the negation test.

 * a. She did a report on pandering politicians.

 * b. *Did she a report on pandering politicians? - Light verb did fails the inversion test.

 * c. *She did not a report on pandering politicians. - Light verb did fails the negation test.

(In some cases, though, have may undergo auxiliary-type inversion and negation even when it is not used as an auxiliary verb – see .) Sometimes the distinction between auxiliary verbs and light verbs is overlooked or confused. Certain verbs (e.g., used to, have to, etc.) may be judged as light verbs by some authors, but as auxiliaries by others.

title: Auxiliary verb


text:
Talk:Ordered pair

Unsorted text
If (x,y) is {{x},{x,y}}, what is (x,x)? -phma

 * Obviously, (x,x) is {{x}, {x,x}} which is the same as {{x},{x}} . But then the expression that determines the second element is wrong.  Suggestions?  --Rade Kutil

 * No, it's quite right. The ordered pair (x,x) is symmetric -- switching the elements has no effect, and that's exactly what is reflected by the set notation. we get (x,x) = { {x} } after full reduction. -- Tarquin

 * Yes, but corresponding to what the article says, x would not be the second element of (x,x), which is wrong. --Rade

 * Then those statements are badly formulated. I've just checked my notes from my university logic class, and it seems we didn't linger much on this. It's not a definition you're meant to work with. The 2nd statement on the main page fails here because it is basically saying "x is the second element IF it's in the pair but not the first element". So if x is both elements, it fails. -- Tarquin

 * No; the second statement doesn't fail for (x,x) =, and both statements are correct. The second statement says in words "x is the second element IF (i) it's in the pair AND, (ii) if there are distinct elements in p, then x is only in one of them".

 * (exist Y in p : x in Y) and (for all Y1 in p, for all Y2 in p : if Y1 ≠ Y2 then (x not in Y1 or x not in Y2)).

 * Since for (x,x), p consists of a single set, Y = {x}, then possibility described in the second clause can never occur - it is never the case that Y1 ≠ Y2, so the second clause of the above statement is trivially true; just as the statement {for all Y in p: if 6 = 9, then Jimi Hendrix lives} is trivially true. Chas zzz brown 06:49 Nov 20, 2002 (UTC)

 * Yes, it is correct now. It wasn't when this discussion was going on and until AxelBoldt inserted the correct statement. --Rade

Beyond that, though, the definition of (x,y) as {{x},{x,y}} is arbitrary. It's a "dummy definition", if you will, like there are "dummy variables".

 * Hmmm... putting on my Platonic hat, I have to disagree here that this is an arbitrary "dummy" definition. It's a definition, and a valid one at that; no different than, for example, the definition of ∑undefined{f(i)} as shorthand for f(1) + f(2) + f(3) + f(4) + f(5). As noted, it's rather clumsy to write in set notation; and thus the usual ordered pair is both clearer and more natural to work with; but I sleep better at night knowing that there is a consistent definition underlying this shorthand! Chas zzz brown 06:49 Nov 20, 2002 (UTC)

It serves only to give a solid set theory foundation to what we do with ordered pairs -- once we've seen it works we can ignore it. Analogously, complex numbers are initially defined as ordered pairs, and once we introduce the notation i = (0,1), we can cleanly forget about the ordered pair and write x+iy -- Tarquin

Why not use the definition (x,y) as {x,{x,y}} ? It could avoid the confusion when x=y. -- Wshun

 * This then requires the axiom of regularity (a.k.a. axiom of foundation) to handle the possibility that one has sets x and z, with x={z} and z={x}, but not x = z. Then (x,x) = {x, {x, x}} = {x, {x}} = {x, z} = {{z}, z} = {z, {z, z}} = (z,z). Bzzzzt!

 * If one assumes the axiom of foundation (as usual in ZF), then x = is forbidden (sets then can't have themselves as members); and then your definition works. The advantage of (x,y) = {{x}, {x, y}} is that it is more general, as it still works without foundation; so it can even be applied to classes as well as sets, etc.. While there may be some confusion when x=y, the definition still works. Chas zzz brown 22:40 Feb 22, 2003 (UTC)

 * If (a,b) = (c,d) and we are trying to prove that a=c and b=d (which is the main requirement for a definition of ordered pair), then {a,{a,b}} = {c,{c,d}}. Then either a=c or a= {c,d}.  If a=c then {a,b}={a,d}, so that b=d and we are done with the first case.  The second case; if a= {c,d} then {a,b} = c.  Then there is an infinite recursion: a= {c,d} = {{a,b},d} = {{{c,d},b},d} etc. and then the proof that a=c, b=d cannot be completed.  Well, it could still be possible, if a=c then {a,b} = {c,d}, {a,b} = {a,d}, then b=d, QED.  But there is no guarantee that a=c in this second case, is there? (NO QED)

On the other hand, letting (a,b) = {{a},{a,b}}, if (a,b) = (c,d) then {a} = {c} OR {a} = {c,d}. If {a} = {c} then a=c. If {a} = {c,d} then card {a} = 1 = card {c,d} so c=d; {a} = {c,c} = {c}, therefore a=c in both cases. Then {{a},{a,b}} = {{a},{a,d}}, thus {a,b}={a,d}, b=d. QED.

-- AugPi

 * Here's a proof written out on Metamath: http://us.metamath.org/mpegif/opthreg.html -- Sekkusu —Preceding unsigned comment added by 67.201.195.107 (talk) 22:02, 29 January 2008 (UTC)

If it is required that card (x,y) = 2 even when x=y or there is any dependence between x and y, define (x,y) as
 * $$ (x,y) = \{\{\,0,x\},\{\,1,x,\{y\}\}\} $$

Then to define an ordered triple so that card (a,b,c) = 3, let (a,b,c) = { (0,a), (1,b), (2,c)} where 0,1, and 2 are Von Neumann cardinals: 0 = {}, 1= –, 2={{}, – }. Likewise, for an n-tuple, let (a1, a2, ..., an) = {(0,a1), (1,a2), ..., (n-1,an)}, then the cardinality of such n-tuple will be n. -- AugPi

Extrapolating the tuple sequence
Can the sequence triple, quadruple, etc. be extrapolated?? Let me see:

 * 2. pair
 * 3. triple
 * 4. quadruple
 * 5. quintuple
 * 6. sextuple
 * 7. septuple
 * 8. octuple
 * 9. nonuple
 * 10. decuple
 * 11. undecuple
 * 12. duodecuple
 * 13. tredecuple
 * 14. quattuordecuple
 * 15. quindecuple
 * 16. sexdecuple
 * 17. septendecuple
 * 18. octodecuple
 * 19. novemdecuple
 * 20. vigintuple
 * 30. trigintuple
 * 40. quadrigintuple
 * 50. quinquagintuple
 * 60. sexagintuple
 * 70. septuagintuple
 * 80. octogintuple
 * 90. nonagintuple
 * 100. centuple
 * 125. quasquicentuple
 * 150. sesquicentuple
 * 175. terquasquicentuple
 * 200. bicentuple
 * 300. tercentuple
 * 400. quatercentuple
 * 500. quincentuple
 * 1000. milluple
 * 10^6. milliontuple

66.245.6.12 00:50, 25 Aug 2004 (UTC)

Other pairs
I replaced the text In the usual Zermelo-Fraenkel formulation of set theory including the axiom of regularity, ordered pairs (a, b) can also be defined as the set { a, { a, b'' }}. However, the axiom of regularity is required, since without it, it is possible to consider sets x and z such that x = {z}, z = {x}, and x ≠ z. Then we have that
 * (x, x) = { x, { x, x }} = { x, { x }} = { x, z } = { z, x } = { z, { z }} = { z, { z, z }} = (z, z)

although we want (x,x) ≠ (z,z).

by a few examples of possible other definitions, plus reasons why they are not used. Aleph4 13:44, 18 Feb 2005 (UTC)

I added a reference to the first definition of the ordered pair using sets alone (due to Norbert Wiener, 1914) and attributed the Rosser pair correctly to Willard van Orman Quine.

Randall Holmes 21:39, 14 December 2005 (UTC)

Wiener definition
Is the Wiener definition actually
 * $$(x,y)=\{\{\{x\},\emptyset\},\{\{y\}\}\}$$?

 * Yes.

Why did he not use
 * $$(x,y)=\{\{\{x\},\emptyset\},\{y\}\}$$? --NeoUrfahraner 06:13, 22 March 2006

 * Because he introduced his definition the the framework of Russell and Whitehead's theory of types. There all elements in a class must be of the same type, namely one less than the type of the class (they are elements of). Now if x and y are of, say, type n, then {x} and {y} would be of type n+1, and {{x}, 0} would be of type n+2. Now putting {{x}, 0} and {y} in the same class would mean a "type clash". Hence we have to use { {y} } --which is also of type n+2-- instead of {y}. —Preceding unsigned comment added by 91.59.209.222 (talk) 00:09, 3 December 2007 (UTC)

definition of caartesian product uses ordered pair, but Morse definition of ordered pair uses cartesian product
in the 2nd paragraph it says:

"The notion of ordered pair is crucial for the definition of Cartesian product"

in the paragraph "morse definition" it then says

$$(x,y)=\{x \times \{0\}\}\cup\{y \times \{1\}\}$$ which uses a cartesian product. so an ordered pair is defined in terms of a cartesian product which is defined in terms of an ordered pair which is defined in terms of a cartesian product etc 12:38, 7 April 2007 (UTC)

Set vs Multiset?
As the Kuratowski definition defines an ordered pair (a,b) as the set (a,b)= {{a}, {a,b}}, what if a=b? Quote: "However, for purposes of foundations of mathematics it has been considered desirable to express the definition of every type of mathematical object in terms of sets". That means the 'set' {a,b} isn't a set but a multiset. Am I correct or just running in circles?

 * If a=b, then $$(a,b)= \{\{a\}\}$$, i.e. if the set defining the pair has only one element, then both components of the ordered pair are equal. I see no need for a multiset. --NeoUrfahraner 10:28, 28 September 2007 (UTC)

Mistake in introduction?
"In mathematics, a pair (a,b) is ordered if (a,b) is not equivalent to (b,a)."

Isn't that only true if it is also true that a is not equivalent to b?- (User) WolfKeeper (Talk) 00:15, 24 February 2008 (UTC)

 * The person who made this edit appeared to be trying to clarify the introduction, which currently seems to be rather technical. The addition doesn't seem to handle the possibility that a and b are the same thing.  I've reverted the edit, since I don't see an obvious way to make the introduction clearer. Michael Slone (talk) 03:26, 24 February 2008 (UTC)

 * I attempted some rewording to the idea clearer and less technical (or at least easier to read...). Although probably more understandable, I'm not sure if it's rigorous enough.  We might want a multi-paragraph lede to explain the concepts in more detail.  —AySz88\ ^ - ^  08:21, 2 March 2008 (UTC)

Another Definition and Proofs
It would be nice to provide a proof for the simple treatment [(x,y)= {x,{x,y}}] as this is the definition given in many introductory math texts. Also, I remember somewhere seeing another definition of the ordered pair (maybe in Quine's Philosophy of Logic?) such that:

The ordered pair (x,y) = {x,{y}}. I'm probably missing something (and perhaps this definition runs into difficulties with iteration) -- but I don't think I saw that definition in the article. Thoughts? —Preceding unsigned comment added by 169.231.34.169 (talk) 02:33, 30 June 2008 (UTC)

Redundancy
Can we delete the "reverse definition"? Its completely identical to the "regular" definition, and the distinction is only syntactic —Preceding unsigned comment added by AbnormallyNormal (talk • contribs) 14:49, 3 September 2008 (UTC)

In mathematics, an ordered pair is a collection of two distinguishable objects
Having the Kuratowski pair <{},{}>, it is totally unclear to me, what does "distinguish" {} and {} mean? Peta 77.104.243.33 (talk) 17:42, 27 March 2009 (UTC)

Wouldn't a changed formulation like:
 * "In mathematics, an ordered pair is a collection of objects having two coordinates (entries, projections), such that one can always uniquely determine the object, which is the first coordinate (first entry, left projection) of the pair as well as the second coordinate (second entry, right projection). If the first coordinate is a and the second is b, the usual notation for an ordered pair is (a, b). The pair is "ordered" in that (a, b) differs from (b, a) unless a = b."

better communicate, that the coordinates of the pair need not be "distinguishable"? Peta 77.104.243.33 (talk) 09:28, 30 March 2009 (UTC)

 * The first sentence is a description of a Cartesian plane geometry, not of an ordered pair; a pair is certainly not a "collection" of pairs. The language here also seems to say that the object is the first coordinate, and the role of the second is left unclear.
 * Nor is geometry the only application of ordered pairs.
 * My attempt (five months ago) to improve this muddle was reverted today with this note: "distinguished" was incorrect, both objects can be the same, i.e. indistinguishable. So what?  If two coordinates happen to have equal values, they are still distinguished by their function.  In the pair (2,2), is there any doubt which is the first coordinate and which is the second?  —Tamfang (talk) 18:38, 18 November 2010 (UTC)

 * My notes and reasons why I reverted your formulation:

 * 1) In your formulation you replaced "collection of objects" by "combination of objects". That is an error, since according to Combination_(disambiguation) "In mathematics, a combination is a collection of things in no specific order.", while in ordered pair, the order is what matters.
 * 2) In your formulation you replaced "collection of objects" by "combination of two objects". That is an error, since the "two objects" can in fact be just one object.
 * 3) In your formulation you wrote that the "two objects" were "distinguishable". That is an error, since as mentioned, there may not be "two objects", in which case, there are no "two objects" to distinguish. To notice, how problematic such a formulation is, notice, that the most commonly used (Kuratowski) definition, gives us $$\left\{ \left\{ \emptyset \right\} \right\}$$ as an example of an ordered pair. Where you have your "two objects", is a mystery.
 * 4) Certainly, the coordinates are always distinguishable, in the sense, that you can tell which object is the first coordinate, as well as tell, which object is the second coordinate, but that is a different sense than the one you used in your formulation.Ladislav Mecir (talk)

 * Good point about combination; but doesn't the same objection apply to collection? — How about structure? or an object consisting of two elements?
 * The most natural reading of "a collection of objects having two coordinates" is something with two levels of structure, viz a set of ordered pairs; otherwise either objects or collection is redundant.
 * Thank you for pointing out Kuratowski's pathological example. (In $$\left\{ \left\{ \emptyset \right\} \right\}$$, how do you tell which coordinate is which?)  You see it as disallowing the obvious operational definition; I see it as a good reason to avoid torturing oneself by hunting for essences.
 * That's exactly the sense of distinguished that I used. Did you confuse it with distinct (i.e. 'unequal')? —Tamfang (talk) 23:58, 18 November 2010 (UTC)

 * A "collection" is certainly a more "basic" notion, than a "combination", as is demonstrated by the fact, that a combination is defined as a collection of a specific kind.
 * I tend to agree, that the word "objects" looks redundant in the present formulation, so would it be better if we wrote: "In mathematics, an ordered pair is a collection (of objects) having two coordinates (or entries or projections), such that it is distinguishable, which object is the first coordinate (or first entry or left projection) of the pair and which object is the second coordinate (or second entry or right projection) of the pair." Would that be an improvement?
 * Re "how do you tell which coordinate is which?" - it is easy, the first coordinate is the $$\emptyset$$, which happens to be the second coordinate as well in this specific case.Ladislav Mecir (talk) 00:29, 19 November 2010 (UTC)
 * ,i.e. we are able to "distinguish" the coordinates in the sense, that we can "distinguish", which object is the first coordinate (the $$\emptyset$$), as well as "distinguish" which object is the second coordinate (the same $$\emptyset$$), while we certainly cannot distinguish $$\emptyset$$ from $$\emptyset$$. Ladislav Mecir (talk) 11:05, 20 November 2010 (UTC)
 * In fact, the "Kuratowski pathological example" cannot be attributed to Kuratowski, since the ordered pair (a, a) has this property for any definition used. - Let's just for a moment suppose, that a denotes you. It certainly isn't possible to distinguish you from you, while it certainly is possible to distinguish, which object is the first coordinate of such a pair (you) as well as which object is the second coordinate of the said pair (you again).Ladislav Mecir (talk) 00:47, 19 November 2010 (UTC)
 * It certainly is possible to distinguish two representations of me from each other. —Tamfang (talk) 23:06, 19 November 2010 (UTC)

Is "99" a one-digit number? —Tamfang (talk) 05:54, 19 November 2010 (UTC)
 * once again, your formulation "two objects ... the first ... is distinguished from the second" is the formulation you wrote, and it is an error, since that is not true. That is like saying that in "99", the first "9" is distinguished from the second "9", while it is not, they are both the same character. What can be distinguished is not "the first from the second", which can not be distinguished, but "what is the first" (between all conceivable objects) and "what is the second" (between all conceivable objects). The fact, that both can be the same clearly disallows to distinguish the first from the second.Ladislav Mecir (talk) 08:16, 19 November 2010 (UTC)

 * Ordinary people would say "99" consists of two nines, a first "9" and a second "9", distinguished by their position. You insist that two congruent objects are the same object.  Such an axiom may be useful for some purposes, but I doubt there's a consensus for applying it universally.  It makes your definition obscure, with no apparent benefit.
 * Could you tolerate a plainer definition (before or after yours) if it were labelled as "informal"? —Tamfang (talk) 20:30, 19 November 2010 (UTC)

 * Your formulation: 'first "9" and second "9" distinguished by their position' is just an error. It requires all objects to have a position in every pair as their own property. But, e.g. an empty set, or many other mathematical objects simply do not have such a property. In your error, an ordered pair having you as its first as well as second coordinate cannot exist. I can feel sorry for you, but cannot allow you to write such falsities into an encyclopedia article. I cannot help but wonder, how people making more than three serious errors in one sentence can feel to be "chosen" to correct it.Ladislav Mecir (talk) 21:34, 19 November 2010 (UTC)
 * "You insist that two congruent objects are the same object." - error again. I just insist, that there are such ordered pairs that contain only one object, which plays the role of both their first as well as second coordinate, and that this property is essential for ordered pairs to be what they are.Ladislav Mecir (talk) 21:57, 19 November 2010 (UTC)

 * Your argument is narrow-minded, if not circular, and your tone is increasingly offensive.
 * "It requires all objects ... as their own property": No, it requires the representations of the objects to have an attribute which we may call context.
 * I chose myself to propose language that might make sense to the layman, just as you chose yourself to defend the true religion of obscure mumbo-jumbo. Any other self-nominees are welcome to put their oar in. —Tamfang (talk) 23:10, 19 November 2010 (UTC)
 * If mentioning one dollar in two different contexts were capable of making two distinguishable dollars from it, I would have been using such a "device" for years.Ladislav Mecir (talk) 10:58, 20 November 2010 (UTC)
 * Thanks for demonstrating the limits of your understanding. —Tamfang (talk) 17:25, 20 November 2010 (UTC)

 * Maybe this will help: what is distinguishable, is, which formulation (context, or projection) we use to refer to the object, i.e. whether we say, that "the object is the first coordinate of the pair", or "the object is the second coordinate of the pair". What is not distinguishable, is the object being the first coordinate of the pair, from itself, being the second coordinate of the pair as well.Ladislav Mecir (talk) 11:27, 20 November 2010 (UTC)

Recent article by Dana Scott and McCarty
I wish to draw everyone's attention to the following article:

Dana Scott and McCarty, Dominic (2008) "Reconsidering Ordered Pairs," Bulletin of Symbolic Logic 14(3): 379-97.

The entry should mention this article in some way, and incorporate a bit of its rich content.111.69.249.90 (talk) 12:42, 25 October 2010 (UTC)

Math formatting
There is some mixture of math formatting on this article. For simple ASCII math (anything with just letters, punctuation, and subscript/superscripts), I propose we just use HTML formatting and lose all the "math" tags, many of which are showing up as images for me. — Carl (CBM · talk) 13:58, 4 April 2011 (UTC)
 * Agree. HTML is preferred where  possible. Paul August ☎ 14:21, 4 April 2011 (UTC)
 * I also agree. There is nothing on this page that requires math tags, and that is not likely to change. Hans Adler 14:24, 4 April 2011 (UTC)

intro
I'd like to nominate the first sentence of this article as the most obscure definition of the most elementary concept. Really, someone who's mathematical knowledge is sufficiently slight that they don't know what "ordered pair" means will not have a chance of understanding the first sentence.

Having said that, I'll try to fix it. McKay (talk) 06:44, 2 September 2011 (UTC)

"Fix"? It contains just your original research "a list of two things" - Where exactly did you find that an ordered pair is a "list"? Where exactly did you find that ordered pair shall contain "things"? Where exactly did you find that ordered pair has to contain "two" "things"?81.200.57.149 (talk) 00:29, 13 October 2011 (UTC)

Ackermann set theory + proper classes
Does it make sense to use the Kuratowski ordered pair definition for proper classes in Ackermann Set Theory resp. its extension ARC? If so, shouldn't this also be mentioned? -- 132.231.198.38 (talk) 15:32, 22 June 2012 (UTC)

Please "spell out" Kuratowski
Could someone _please_ do us the favor of spelling out how exactly these definitions, let's pick Kuratowski since it's so common, actually "define" an ordered pair?

Kuratowski says: (a,b) := {{a}, {a,b}}

Let's suppose a and b are very simple objects, say 6 and 13. So from Kuratowski we can supposedly get to the ordered pair (6,13) from this:

{{6}, {6,13}}

... so at the "top level" this is a set that contains two sets, in no particular order. How is this equivalent to a pair which is ordered and contains integers? Clearly I, for one, am missing something here. Either there is more to the notation, or something more is being implied. Thanks for any enlightenment! Gwideman (talk) 11:59, 25 August 2012 (UTC)

 * It's pretty clear that all by itself, just from the symbol string {{a}, {a,b}}, a mechanism (e.g. a Turing machine) cannot determine the first element and then the the second element and then properly write the symbol string (a, b). The TM will need a set of rules embedded in its "program" to guide it. (Intuitively for folks who like puzzles, it should be suspicious that there's a singleton {a} and then a pair {a, b} and the element in the singleton, a, appears in as an element in the pair {a, b}.) By the time Suppes 1970:22-23 introduces the ordered pair (Kuratowski version) he has introduced 5 axioms of set theory:
 * 1 Axiom of extension (set A equals set B if and only if they have the same elements)
 * 2 Axiom of specification (a "condition" S(x) creates a set B from the set A for all elements of A for which S(x) holds)
 * 3 Axiom of pairing (creates unordered pairs)
 * 4 Axiom of union (combining sets)
 * 5 Axiom of powers (power set E contains all subsets of the given set)

 * So by the time we reach his Section 6 Ordered Pairs there's powerful set of rules available to define the notion of ordering. By use of the definition (a, b) = {{a}, {a,b}}, Suppes' proof proceeds by cases and it has to do both the if and the only-if parts, so unfortunately it is rather long (about 200 words) and somewhat tedious. It begins with the statement "We have to show that if (a, b) and (x, y) are ordered pairs and if (a, b) = (x, y), then a = x and b = y." It proceeds through all the various singleton cases, then arrives at the x <> y case.
 * Maybe someone knows a shorter proof somewhere. Wvbailey (talk) 16:22, 25 August 2012 (UTC)Bill

 * Wvbailey: Thanks for your response. Your "Turing machine" comment succinctly captures where I was stuck, and indeed the "program to guide it" is instrumental in understanding this ostensibly simple piece of set theory.

 * The crux of what stumped me (and evidently was what Peta and Tamfang discussed earlier on this talk page) is this: (a,b) := {{a}, {a,b}} does *not* define a pair with an ordering per se (in the sense of identifying that a is first and b is second). It only defines a way to use the mechanism of sets which distinguish items only by value, to implement two distinct variables -- that is to say, two distinct slots in which to store items, and later retrieve the items from each distinct slot.

 * The "distinct slot" idea is more or less just a restatement of the "defining property" of the ordered pair, and a reader who neglects to note that the defining property of "ordered pair" omits any mention of "order" will be vulnerable to confusion downstream.  It remains for additional "logic" to actually perform the retrieving of one item or the other, and attribute to them the order (first and second).  And thanks *very* much to the author who explicitly included that logic  (Pi1 and Pi2) that actually performs that job.  That crucial piece was missing from Hrbacek "Introduction to Set Theory", which is how I got here in the first place.

 * For the sake of others who might be similarly stuck: In order to understand how "{{a}, {a,b}}" corresponds to "ordered pair", it's crucial to understand how "first item" and "second item" are picked out of "{{a}, {a,b}}".  A key challenge is that "{{a}, {a,b}}" has no ordering to it, so, for example, the {a} element is just an element in the set, not the first element per se. Indeed, so far as I can see, Karatowsk could just as well be written: (a,b) := {{a,b}, {a}}  (not to be confused with (a,b)reverse, mentioned later in the article).

 * The answer lies in the two Pi equations, which in English are:

 * first: take the element that appears in both members.
 * second: take the element that appears in only one of the members. If there is none, the elements must be the same, so use that.

 * Hope that this is all correct, and might help other folks who stumble in here! Gwideman (talk) 00:51, 26 August 2012 (UTC)

Couple of questions/suggestions
1. Intro: "and hence the ubiquitous functions". Here "ubiquitous" looks suspiciously like a technical word. Is there some distinction between ubiquitous functions and non-ubiquitous functions? Or is this simply pointing out that functions are of broad importance?

2. In "Defining the ordered pair using set theory": "The above characteristic property of ordered pairs is all that is required to understand the role of ordered pairs in mathematics. Hence the ordered pair can be taken as a primitive notion, whose associated axiom is the characteristic property."

The first sentence here seems fatuous. Even if true, the second sentence doesn't follow from it.

I suspect this is trying to say something like: the ordered pair is a powerful concept, and consequently its axiom (the characteristic property) has been chosen as a primitive upon which to build further set theory.

Gwideman (talk) 01:08, 26 August 2012 (UTC)

 * Re your first question: your second interpretation is correct, the formulation is simply pointing out that functions are of broad importance.
 * Re your second question: I did not write the formulation, however, the first sentence is OK, and the second sentence follows from it. It is not true that the first sentence is trying to say that "the ordered pair is a powerful concept", rather it says that it is easy to characterize the notion of the ordered pair, since the above characteristic property is all that is needed. It is also not true that the ordered pair has been chosen as primitive and its characteristic property has been selected as an axiom, at least not in general. Commonly, instead of chosing the notion of the ordered pair as primitive, set theories define the ordered pair using some definition such as Kuratowski's. Ladislav Mecir (talk) 18:52, 17 May 2016 (UTC)

Coordinate extraction formulas
In the formulas given under Kuratowski definition for extracting the first and second coordinates, what is the purpose of the outermost Union? I had difficulty finding a definition of what unary union does in all cases. Is the idea to "remove an outer layer of set enclosure"? Ie: {a} --> a? But if a is not a set, but a value such as 6, will this succeed? I expected the result of a union operation to be a set. Gwideman (talk) 13:02, 28 August 2012 (UTC)

 * Everything, including "6", that can be put in a set is a set. --JBL (talk) 13:45, 28 August 2012 (UTC)

Ambiguity
The notation {(a, b)} is ambiguous. It can mean either the singleton whose single element is (a, b) or the doubleton whose elements are (a and b). GeoffreyT2000 (talk) 02:45, 18 June 2015 (UTC)

Contradiction with Tarski–Grothendieck set theory article
The article says: "Also note that the short version is used in Tarski–Grothendieck set theory, upon which the Mizar system is founded." However, under the Mizar implementation of Tarski–Grothendieck set theory, it shows that the Kuratowski definition is used. This is a contradiction, so one of these articles must be wrong. 173.76.101.86 (talk) 02:04, 7 July 2016 (UTC)

Mathematical objects?
Although it has been in the first sentence since 2011, calling the components of an ordered pair mathematical objects is clearly false. The ordered pair is certainly a mathematical object (a mathematical structure) but its components need not have any mathematical meaning (they could be colors or the names of cities or anything else with no mathematical import). I am amazed that editors have let this stand and am only bringing this up here before I change it on the off chance that someone may want to defend it. Bill Cherowitzo (talk) 03:02, 15 September 2016 (UTC)

 * "calling the components of an ordered pair mathematical objects is clearly false" - I am sure this is your honest opinion, but, please consult how Wikipedia handles these issues. If you do not have any WP:IS stating that the components of the ordered pair are not mathematical objects, please do not make the change. Ladislav Mecir (talk) 04:00, 15 September 2016 (UTC)

 * Do you actually disagree substantively? --JBL (talk) 12:41, 15 September 2016 (UTC)

 * Yes, I disagree sustantively too. Definitions (e.g. Kuratowski's definition) of ordered pair are restricted to pairs of sets, which are mathematical objects. There are also definitions of ordered pairs of classes, but that does not matter in this case, since classes are mathematical objects too. Ladislav Mecir (talk) 14:17, 15 September 2016 (UTC)

 * Actually I think the burden of proof is not for me to find a source saying that the components are not mathematical but rather for you to find one that says that they are. Appealing to any of the various definitions in terms of sets is a formalism that does not cut the mustard! To say that since a formal definition involves sets, the components are mathematical, is begging the question. The second sentence refers to the order of the objects (components), yet these are not the sets of the definitions, and my point is that these objects need not be mathematical. Consider a reader coming to this page for the first time. In order to understand the first sentence the reader must already be familiar with the formal definitions and their consequences. This is simply poor writing and one of the major contributors to the oft heard complaint that math articles (in particular) are written by and for specialists. This is a topic that is taught with varying degrees of rigor and across several disciplines. To treat it solely as a topic in formal logic is doing a disservice to its varied audiences. Following the best traditions of Wikipedia technical writing, the lead should be an informal introduction to the topic and, in this case, explain the need for a more formal definition before getting into the mechanics of those definitions. The present article does a poor job of this and the given rationale for the set-theoretic definitions is, IMO pure BS (and, might I add, unsourced). This article's introductory material does need some work, and I think that a good start would be the removal of the word mathematical from the first sentence. --Bill Cherowitzo (talk) 19:19, 15 September 2016 (UTC)
 * I agree with Bill. Certainly the components of an ordered pair need not be any more "mathematical" than the elements of a set, which in non-formal settings are usually allowed to be anything. Paul August ☎
 * I am more partial to the formalism than either Bill or Paul (in particular, the idea that a set can contain something that is not mathematical makes me a little queasy), but I agree that it is not necessary in the lead. I don't know what I'd do with the wikilink mathematical object.  --JBL (talk) 23:51, 15 September 2016 (UTC)
 * As a modern mathematician, raised on formalism, that idea would make me feel queasy too (like walking in quicksand) while doing mathematics, but not in other settings. And besides mathematics is bigger than formal mathematics. Paul August ☎ 12:25, 16 September 2016 (UTC)
 * Formalism is my bread and butter as well, but I am wondering if I am missing something here since I don't feel the slightest bit of queasiness. How does one fill out the sentence, "An element of a set is mathematical because, ..." – it is an element of a set???? That kind of circularity would make everything mathematical and so, that term would lose all meaning. I have been accused in the past of not seeing the forest because of the trees, and if I am doing that again I would like to be set on the straight path to queasiness. --Bill Cherowitzo (talk) 17:06, 16 September 2016 (UTC)
 * [Disclaimer: the following comments are not intended to have anything to do with editing this article.] In my opinion, the formalism of "set" can contain only mathematical objects, by definition.  I don't think it makes sense to speak of a set containing a (real) tree, any more than it makes sense to speak of a tree that is Ω meters tall.  In the real world there are collections of things, but they are not sets.  (I do not think that it is important that other people agree with me, nor do I think it is necessary to share my view to know how to write this article well.  I also do not claim that my feeling about this is "right" in any strong sense; it is just my feeling.)  --JBL (talk) 18:51, 16 September 2016 (UTC)

 * I mainly disagree with the formulation "calling the components of an ordered pair mathematical objects is clearly false". I think that the main cause of the controversy is the fact that the term mathematical object is from the philosophy of mathematics. Mathematicians prefer the term object, and for them a formulation calling an element of an ordered pair object is fine. So, for the consensus sake, I can agree with replacing the term mathematical objects by objects in the lead section of the article. Ladislav Mecir (talk) 07:03, 17 September 2016 (UTC)
 * Perhaps I should have qualified my statement by adding "... from certain perspectives." ( such as my own!) I think this change will improve the tenor of the article. I would also like to see some more material having an informal perspective, but I do not want to dilute any of the fine work presented here on the various formal definitions. Perhaps a section that is clearly labelled as being informal and other dealing with applications can be added. Bill Cherowitzo (talk) 22:25, 17 September 2016 (UTC)

Boldface in the "Generalities" section
The section contains this text: "Then the characteristic (or defining) property of the ordered pair is:" Per BOLD, the use of boldface for this purpose is discouraged. I propose to use "Then the characteristic (or defining) property of the ordered pair is:" instead. Ladislav Mecir (talk) 04:21, 21 September 2016 (UTC)
 * ✅ I agree. --Bill Cherowitzo (talk) 18:03, 21 September 2016 (UTC)

Kuratowski "01" variant
In the Variants section of Kuratowsi's definition, the following is listed: $$(a,b)_{01} = \{\{a,0\},\{1,b\}\}$$. It seems like this definition would not be adequate, as we would have for example $$(\{0,1\},b)_{01} = (\{1\},b)_{01}$$. Does this pair require a custom definition of equality, or what? — Preceding unsigned comment added by Luqui (talk • contribs) 16:47, 29 April 2018 (UTC)

Gen math
Ordered pair rectangular coordinate plane and quadrant 114.108.244.56 (talk) 20:46, 11 September 2022 (UTC)

 * ✅ - I guess, you suggest to add some remarks about using ordered pairs in analytic geometry. More generally, we should have a section about typical applications of ordered pairs. Currently, the article only explains how ordered pairs can be specified and implemented, but not what they are good for. - Jochen Burghardt (talk) 07:52, 12 September 2022 (UTC)

title: Talk:Ordered pair


text:
Richard Lloyd (guitarist)

Richard Lloyd (born October 25, 1951) is an American guitarist and songwriter, best known as a founding member of the rock band Television.

Early life
Lloyd first became interested in music as a child.

Lloyd attended Stuyvesant High School in New York City. In his early teens, he studied drums with William Kessler, who was the ghostwriter for Cozy Cole, one of the famous big-band drummers. A few years later, he turned to the guitar.

Teenage years
When Lloyd was in his middle teens, he met a fellow guitarist from Brooklyn named Velvert Turner. Turner claimed he knew Jimi Hendrix. Per Turner, Hendrix considered Turner his "little brother", and took him on as his protégé, inviting him to various clubs and teaching him guitar from Hendrix's apartment on W 12th St. As Turner and Lloyd were best friends, Turner asked for permission to teach Lloyd what he was learning, and so Lloyd and Turner began practicing together under the teaching auspice of Hendrix. Lloyd frequently attended Hendrix's shows, as well as those of other well-known acts such as Led Zeppelin, The Who, The Jeff Beck Group with Rod Stewart, The Allman Brothers Band, and the Grateful Dead.

At Stuyvesant High School, during the 11th grade, Lloyd decided that he was going to become a well-known guitarist, and so he stopped bringing his textbooks or homework to school, instead bringing an electric guitar in a hardshell case. When his teachers would ask him where his schoolbooks were, he would point to the case. Being asked to open the case, the teachers would proclaim "I don't see any books in there – only an electric guitar", and Lloyd's retort was "That's the book I'm studying." He still managed to pass all of his tests but quit in May 1969, before graduation, because he did not believe in diplomas, and he considered that having a high school diploma would not do him any good in his pursuit of the music business. That summer his parents moved to Montclair, New Jersey, and gave him an ultimatum: he could either get a job or repeat the 12th grade in a new high school. He chose the latter, which led to his deep friendship with another guitarist named Al Anderson, who went on to be one of a couple of Americans to play in Bob Marley's touring band, The Wailers, along with Donald Kinsey.

Early music career
After quitting a second high school before graduation, he moved to Boston, where he lived for two years. It was during this time that he enjoyed his first public performance, sitting in with John Lee Hooker at the Jazz workshop on Boylston Street. By this time Lloyd had already met and knew Jimi Hendrix, Buddy Guy, and members of Led Zeppelin, and could be frequently found backstage conversing with his heroes. John Lee Hooker asked him what he did and Lloyd replied that he was a guitarist, whereupon John Lee Hooker called him over and whispered into his ear "the secret of the electric guitar" and invited him to sit in with the band. After living in Boston for a period of time, Lloyd moved back to New York, but desired to visit other centers of musical culture. He first decided to hitchhike across Route 66, but after considering how long it might take, he flew directly to Los Angeles instead. He spent the next two years in Los Angeles from 1971, practicing the electric guitar without an amplifier and going to record company parties, which at that time were very ostentatious. He lived as a roommate with Richard Cromelin, the music critic for the Los Angeles Times, so again, he had access to the highest level and echelon of rock 'n roll.

In 1973, he heard about the New York Dolls and the beginning of a brand-new New York scene, so he arranged with a friend who owned a Lotus Europa to take him as a passenger across the country. As they made their way through the country, with stops in famous towns like Memphis and New Orleans, he reached New York and heard the unpleasant news that the Mercer Arts Center, known for hosting the first concerts of bands like the Dolls, the Modern Lovers and Suicide, had collapsed. He began frequenting shows at the Diplomat Hotel and other spots, and seeing the possibility of a new scene, began contemplating putting a band together. He met a fellow named Terry Ork at Max's Kansas City, who had a huge loft in Chinatown and needed a roommate, so Lloyd moved into Ork's Chinatown loft, living in the front room, a small room facing East Broadway. Ork worked for Andy Warhol making silkscreen prints at night and working at a theatrical poster shop called Cinemabilia during the daytime. Ork very much wanted to manage or create a band such as Warhol had done with The Velvet Underground. During discussions with Lloyd, Ork mentioned that he knew of another guitarist without a band who was auditioning a couple of songs at Reno Sweeney's audition night. Reno Sweeney's was an off-Broadway-style club on W. 13th St that mostly hosted singers like Peter Allen and Bette Midler.

Television
Ork and Lloyd went to Reno Sweeney's one night during the summer of 1973, where Richard saw Tom Miller play three songs. Lloyd leaned over to Ork during the second song and told him that this fellow had something, but was missing something, and what he was missing, Lloyd had. He advised Ork that if Terry could convince Tom, the combination of Lloyd and Miller would have the makings of the band Terry Ork was looking for. This was the beginning of the formation of the band Television. Miller would eventually change his last name to Verlaine and Richard Meyers became Richard Hell and promised to learn the bass as they went along. With the addition of Billy Ficca on drums, the quartet was complete. Television rehearsed seven days a week for five or six hours a day during the fall and winter of 1973, and made their first public performance on March 2, 1974, at the Town House Theatre on W. 44th St.

Television were looking for a club where they could develop an audience and play more often as the house band, when Verlaine spotted a guy putting up the awning on a bar on the Bowery which stood under a flophouse for homeless alcoholics. Verlaine and Lloyd went back up and discussed the possibility of playing in this new club, which was to be called CBGB. After Television's manager Ork promised CBGB a large take at the bar, Television was given a gig at the end of March 1974. CBGB was run by Hilly Kristal, who had planned to feature country, bluegrass and blues (CBGB) at the club, but when several original rock bands like Blondie, the Ramones and Talking Heads started to show up after finding out that there was a place to play, Ork became the official booking agent for the club. CBGB started to get noticed after bands like Television and Talking Heads started to fill the place up, and when a young poet named Patti Smith began playing double bills with Television, the club started becoming famous. CBGB closed its doors in New York in 2007.

After recording demos for various record companies, Richard Hell left the band and was replaced by Blondie's bass player at the time, Fred Smith. Fred Smith's solid bass playing allowed for a more transcendent and profound music from the two guitarists and drummer, resulting in their being signed to Elektra Records in 1977.

Television recorded two albums for Elektra: Marquee Moon and Adventure. As a debut release in 1977, Marquee Moon remains on lists of greatest albums in rock and roll history, and never has been out of print.

After recording Adventure in 1978, and finding success elusive in the United States, Television disbanded after a successful series of dates at New York's The Bottom Line. The various members went their separate ways, although all of them continued in the music industry, occasionally to reunite.

Solo career
After Television first disbanded in 1978, Lloyd released a solo album for Elektra Records, titled Alchemy. His solo career has included work as a guitarist and session musician for various artists including Matthew Sweet, John Doe (of the band X), and Swiss singer-songwriter Stephan Eicher. In 1990, Lloyd recorded a version of the Roky Erikson/13th Floor Elevators song "Fire Engine", which was included on the tribute album compilation Where the Pyramid Meets the Eye.

He is known to teach other aspiring guitarists in his studio in New York City, where he frequently produces other acts. He acted as producer, recording engineer and band member on the Rocket from the Tombs album Rocket Redux, released on Smog Veil Records. He has also produced a number of records for other New York bands, including The Blondes, Inc., Chris Purdy, Miss Mother USA and Kevin Otto; as well as Skinny Girls Are Trouble (2010), the third album by country-punk songwriter Jim Neversink. His other current projects include a reformation of Rocket from the Tombs. He still owns the famous pinstriped Stratocaster he played with Television.

In 1991 he contributed guitar parts to Girlfriend, the album by American alternative rock musician Matthew Sweet.

In 1994, he was cast by Cynthia Plaster Caster.

As of 2016, Lloyd has released seven solo albums, beginning with Alchemy in 1979, Field of Fire in 1986 (reissued in 2007 with an additional disc of music), the live disc Real Time in 1987 (recorded at CBGB) and The Cover Doesn't Matter in 2001. The Radiant Monkey was released on Parasol Records in late 2007. The CD features Lloyd on everything but drums (played by session drummer Chris Purdy, with Television's Billy Ficca on one track). The Jamie Neverts Story, released on Parasol in 2009, was Lloyd's tribute to his hero, Jimi Hendrix; and was dedicated to the memory of his friend Velvert Turner; "Jamie Neverts" was the secret name that Velvert and Lloyd came up with to prevent all the kids in the neighborhood from trying to come along when they went to see Hendrix.

In 2010, he released Lodestones, sometimes referred to as Nuggets From the Vault. It was issued on vinyl in 2018 for Record Store Day. Of the album, Lloyd said: "I found a large box of unmarked cassettes going back quite a few years. When I began listening to them, I found a number of finished products – songs that for some reason I [had] failed to put on records. [I] got a machine to transfer the cassettes to CDs and I transferred [the] CDs to my machine at my studio and mastered them. When finished I had a 10 song record.". According to Lloyd's website, other players on the record are John Klages and David Leonard on rhythm guitar, and most likely Steve Cohen on bass and Ed Shockley on drums. In 2016, he released Rosedale. While Lloyd plays all the instruments on most of the album, Billy Ficca and Chris Frantz also play on a few tracks, and local musicians bassist Terry Clouse and drummer Jeff Brakebill from his new hometown of Chattanooga, Tennessee, are also featured. In 2018, Lloyd published his memoir, Everything Is Combustible: Television, CBGB's and Five Decades of Rock and Roll.

Personal life
In 2015, Lloyd married Sheila O'Keefe. His memoir notes he has a son, Dylan.

Solo discography

 * Alchemy (Elektra 1979)
 * Field of Fire (Moving Target/Celluloid 1986) – later released on CD by Reaction Recordings (c/o Parasol Records)
 * Real Time (1987)
 * The Cover Doesn't Matter (2001)
 * The Radiant Monkey (2007)
 * The Jamie Neverts Story (2009)
 * Lodestones (2010)
 * Rosedale (2016)
 * The Countdown (2018)

title: Richard Lloyd (guitarist)


text:
User:Ewlloyd

Eric W. Lloyd is a software professional with a penchant for acting and voiceover. He is the voice of BodyWorks 6, and has most recently appeared in Tempe (Arizona) Little Theatre's production of Moon Over Buffalo.

title: User:Ewlloyd