text:
Downtown Connector

In Downtown Atlanta, the Downtown Connector or 75/85 (pronounced "seventy-five eighty-five") is the concurrent section of Interstate 75 and Interstate 85 through the core of the city. Beginning at the I-85/Langford Parkway interchange, the Downtown Connector runs generally due north, meeting the west–east I-20 in the middle. Just north of this is the Grady Curve around Grady Memorial Hospital. Continuing north, the terminus of the Downtown Connector is the Brookwood Interchange or Brookwood Split in the Brookwood area of the city. The overall length of the Downtown Connector is approximately 7.5 mi. Since the 2000s, it has been officially named James Wendell George Parkway for most of its length, although it is still designated the Connector in the mainstream. It also has unsigned designations State Route 401 (I-75) and State Route 403 (I-85) along its length, due to I-75 and I-85 having 400-series reference numbers.

Route description
The Downtown Connector carries more than 437,000 vehicles per day at its busiest point — just south of 10th Street in Midtown, while the least traveled portion carries 243,000 vehicles per day — just south of Fulton Street, near the interchange with Interstate 20. The area around the connector and associated interchanges are considered one of the ten-most congested stretches of interstate in the U.S. Due to this fact, many motorists often compare Atlanta to Los Angeles, which is also known for its notoriously-congested freeway system.

The highway is fully instrumented with Intelligent transportation system (ITS) devices. There are more than 25 closed-circuit television cameras between the Langford Parkway interchange (south end) and the Brookwood Interchange (north end). Additionally, the Downtown Connector has three large overhead electronic message signs, and four smaller HOV-dedicated message signs on the median barrier wall. Traffic flow data is gathered through a video detection system, using pole-mounted black-and-white cameras spaced every 1/3 mi on both sides of the roadway. All video and data is fed into the GDOT's Transportation Management Center (TMC), via fiber-optic cable located under the shoulders of the roadway. Virtually all entrance ramps are metered, with the exception of the freeway-to-freeway connection ramps from I-20. As with every other freeway inside the Perimeter, the Downtown Connector is lined with streetlights mounted in the center median, with high-mast lighting at major interchanges.

Atlanta's skyline, both Downtown and Midtown, can be seen from the highway, especially at the northern and southern ends.

Planning and initial construction
On December 14, 1944, the Georgia State Highway Department, the predecessor agency to the Georgia Department of Transportation (GDOT), collaborated with the city of Atlanta, Fulton County, and the Bureau of Public Roads (BPR), the predecessor to the Federal Highway Administration (FHWA) to work on a plan to improve traffic and connectivity within the Atlanta area. Chicago-based H.W. Lochner & Company, a transportation engineering firm which had been formed earlier that year, was hired to provide a comprehensive transportation plan for the region, using data conducted by the BPR and the state highway department from 1936 to 1945, and future growth projections. The report, which was released on January 10, 1946, recommended a total of six radiating expressways be constructed in the city, with the intent of being integrated into what later became the Interstate Highway System, which at the time was expected to eventually be authorized by Congress. The report recommended that four of these expressways converge into a single north-south route through downtown, which was referred to at the time as both the "Downtown Connector" and the "Downtown Connector Expressway".

Work of the first section of the Downtown Connector, located between Williams Street and the Brookwood Interchance, as well as the approximately 1/3 mi stretch of what is now I-85 north to Peachtree Street, began in September 1948, and was dedicated and opened to traffic on September 25, 1951. The 0.6 mi stretch from Piedmont Avenue to Williams Street opened on May 5, 1959. Work on a southern stretch of the Connector, a 1.4 mi section between University Avenue and Richardson Street, began in March 1955, and was dedicated and opened on July 25, 1957. Construction on the final stretch, the 2.3 mi stretch between Richardson Street and Piedmont Avenue, including the interchange with I-20, began on February 26, 1962, and was opened to traffic on September 18, 1964. A ribbon-cutting ceremony for the Downtown Connector, attended by Governor Carl Sanders, FHWA administrator Rex Marion Whitton, and Atlanta mayor Ivan Allen Jr. was held inside the Georgia State Capitol on October 15, 1964.

Initial construction of the highway displaced parts of Techwood Drive and Williams Street in Midtown Atlanta. It also destroyed street grids east and south of downtown, dividing Sweet Auburn in two and the interchange with I-20 leveling the northern part of the Washington-Rawson district. The proposed I-485 was originally planned by the Georgia Department of Transportation to carry some south–north traffic through the eastern side of the city, but most of this was canceled in the 1970s by the then-governor of Georgia, Jimmy Carter. Parts of that road are now I-675 and SR 400.

Reconstruction
The highway was heavily reconstructed during the 1980s as part of GDOT's Freeing the Freeways program to widen Atlanta-area freeways, with most of the Connector's width being doubled from three to six or seven lanes in each direction. The project included work to increase lanes from six to eight on I-20, I-75, I-85, and I-285 ("The Perimeter"), as well as ten lanes on the downtown connector, involving 126 total miles and was phased over 13 years between 1976 and 1988. The improvement campaign also included elimination of sharp curves and grades, left-hand exits, excessive interchanges, and short acceleration/deceleration lanes.

So as to offer a bypass around construction through the center of the city, the perimeter road (I-285) was completed first. The radiating expressways were then upgraded, and the last phase was reconstruction of the depressed sections through downtown Atlanta. By June 1983, some $252 million in discretionary funds had been used to complete most of the highways save for some major interchanges and the downtown section. The eight miles of the downtown section, which includes the 4.4-mile long downtown connector, was the most complicated section of the entire reconstruction. Work was started on it in 1984, and it included redesigning the massive interchange between I-20 and I-75/85 at Memorial Drive where much of the mileage was on structure. The downtown connector was to be widened to ten lanes, and this required quite a bit of right of way acquisition. Many bridges, including the 55 over the connector portion alone, had to be designed and built. In addition to the general-purpose lanes, provisions for high-occupancy vehicle (HOV) lanes and dedicated on-ramps at Williams Street, Piedmont Avenue, and Memorial Drive were built, and were subsequently converted to HOV usage in 1996.

Later history
Between 2000 and 2004, the six-lane wide 17th Street Bridge was constructed over the Brookwood Interchange, connecting Midtown Atlanta with the then-new Atlantic Station development. In 2008, reconstruction of the 14th Street Bridge took place in order to accommodate increased traffic flow and pedestrian amenities. This work also included the construction of two new off-ramps: a southbound ramp to 10th Street, and a northbound ramp to 17th Street. This work was completed on May 28, 2010.

In early January 2010, a section of the highway between 14th and 17th streets developed an unusual problem dubbed "phantom ice" during a prolonged cold wave that kept temperatures below freezing for several days in the Atlanta region. GDOT engineers believe that heavy rain in previous months raised the water table and caused it to seep upward through joints between lanes, where it subsequently froze. Commuter traffic was stopped briefly on at least two days in order to treat it with deicing materials and inspect the freeway to determine the cause.

In January 2016, GDOT started a study of the Downtown Connector. Citing high levels of congestion that occur outside of peak periods, the purpose of the study is to gather data and explore options for reducing congestion. Data collection took place all year in 2016, and the data is being analyzed and options evaluated throughout 2017 and 2018. The study is expected to be completed in 2019. The study has already released their data on their website, including nine "Hot Spots", or dangerous and congested points that could most benefit from design improvements, and a detailed analysis of traffic in the whole metro area and how it relates to traffic on the Downtown Connector.

The Stitch
The Stitch is a proposed engineering project that would see portions of the Downtown Connector capped and covered with public greenspaces. In August 2016, Central Atlanta Progress (CAP) announced plans to cover the Downtown Connector, converting it to a subterranean roadway and building a series of greenspaces on top of the road to cover an area of approximately 14 acres, similar to Klyde Warren Park in Dallas. The proposal was first put forward by CAP as a way to reclaim land area taken by the Downtown Connector.

Similar discussions had been held at CAP as early as 2003 and have included similar proposals for the area near the Georgia State Capitol.

This project was drafted by Jacobs Engineering Group and is called "The Stitch", as one of the intents of the project would be to "stitch" together roads that had been divided with the creation of the Downtown Connector. The study, prepared for CAP by Jacobs Engineering Group at a cost of $100,000, called for a 14 acre stretch of reclaimed land covering a 0.75 mi stretch of the Downtown Connector from Spring Street to Piedmont Avenue in downtown. The study, which compared the project to the Klyde Warren Park in Dallas, said the project would cost approximately $300 million and stimulate approximately $1.1 billion in land development and increased property value. Unlike Klyde Warren Park, The Stitch would involve the creation of multiple parks. The reclaimed land would also allow for the reconnection of Atlanta's street grid, which was interrupted by the construction of the Connector.

Between 2017 and 2018, CAP conducted a year-long viability study on the project that involved Jacobs and John Portman and Associates. In March 2019, the Urban Land Institute (ULI) published a feasibility study on the project, while that same month the estimated cost for the project was stated at $452 million. Advisors from the ULI estimated that the project could take four years of planning and pre-development and six years of construction.

As of 2019, the project was still in the pre-construction phase, with an estimated timeline and budget of ten years and $452 million, respectively.

In 2021, the Midtown Connector Project (MCP) Foundation released a proposal to build a park over 10 blocks of the Midtown portion of the Downtown Connector from North Avenue to 10th Street. The proposal connects that area of the city to the main campus of the Georgia Institute of Technology, which is located on the other side of the Interstate.

Exit list
Exit numbers follow the mileposts along I-75.

title: Downtown Connector


text:
Louis V of France

Louis V (c. 966 or 967 – 22 May 987), also known as Louis the Do-Nothing (Louis le Fainéant), was a king of West Francia from 979 (co-reigning first with his father Lothair until 986) to his early death in 987. During his reign, the nobility essentially ruled the country. Dying childless, Louis V was the last Carolingian monarch in West Francia.

Youth
Louis was born c. 966. He was the eldest son of King Lothair of France, the Carolingian ruler of France, and Queen Emma, daughter of King Lothair II of Italy and Empress Adelaide. Louis was associated to the government by his father in 978 and crowned co-king on 8 June 979 at the Abbey of Saint-Corneille in Compiègne by Archbishop Adalbero of Reims.

Marriage
In 982 at Vieille-Brioude, Haute-Loire, the fifteen-year-old Louis was married to the forty-year-old Adelaide-Blanche of Anjou, sister of Count Geoffrey I and twice a widow from her previous marriages with Count Stephen of Gévaudan and Count Raymond of Toulouse, Prince of Gothia. This union was purely political and arranged by the king – following the advice of Queen Emma and Count Geoffrey I – with the double purpose of restoring the Carolingian royal power in the south of the kingdom, and (according to Richerus) to obtain the support of the local southern lords in his fight against the Robertians. Being related by marriage to two of the most powerful southern comital families of the Kingdom, Lothair believed that he could confront the power of Hugh Capet.

Immediately after their wedding, Louis and Adelaide-Blanche were crowned king and queen of Aquitaine by Adelaide's brother Bishop Guy of Le Puy. From the very beginning, however, the mismatched couple was unable to live peacefully together, not only due to the notorious age difference between them, but (according to Richerus) also because of Louis' reported affairs:

 * [...] They had almost no conjugal love; because Louis had barely reached puberty, and Adelaide was old, there was only incompatibility and disagreements between them. They did not share a common bedroom as they could not bear it; when they had to travel, each took a separate residence, and when they were forced to talk, their conversations were in the open air and were never long, but only lasted for a few words. They lived in this way for two years, until they obtained the divorce for their opposite characters. [...] Louis, who did not have a tutor, indulged himself in all sorts of frivolity due to his young age.

In 984, after two years of childless union (and according to Rodulfus Glaber), Adelaide tricked her young husband into making a visit to Aquitaine, and once there, she left him and returned to her family, marrying shortly thereafter Count William I of Provence:

 * When the young prince reached adolescence, Lothair made him king and appointed him his successor; he also chose for him a princess of Aquitaine as his wife, but soon she perceived that the young man would not inherit the talents of his father. Therefore she decided to separate from her husband; and for this purpose she adroitly convinced him to make a trip to her province of Aquitaine, assuming that her hereditary rights there guaranteed her the possession of the land. Louis, without suspecting the artifice, yielded to the advice of his wife, and went with her. When they were in Aquitaine, she left her husband to join her family.

However, despite being recorded by relative contemporary and later sources (Richerus, Rodulfus Glaber, the Chronicon Andegavensi and the Chronicle of Saint-Maxence, among others), the existence of this marriage was recently challenged by historian Carlrichard Brülh.

Reign
Upon his father's death on 2 March 986, the already-crowned Louis V became the undisputed king of the Franks. At that time, however, there existed in the Frankish court two factions: one led by Archbishop Adalberon of Reims and Queen Emma, who, being strongly influenced by her mother Empress Adelaide, wanted the renewal of friendly relationships with the Ottonian dynasty; the other faction wanted to continue Lothair's policy, and taking advantage of the minority of Emperor Otto III, wanted a policy of expansion to the east and the recovery of Lotharingia. In addition, the young monarch inherited a battle between his father's line of elected kings (which had been interrupted twice by Robertians and once by the Bosonids), and the Ottonian house of Emperor Otto I. As defender of Rome, Otto I had the power to name the clergy in Carolingian territory, and the clergy he had named were not supporting the Carolingians.

Initially, Queen Emma dominated the situation, but in the summer of 986 there was a reversal: the Anti-Ottonian party prevailed, after which she was forced to leave court and seek refuge with Hugh Capet. This event also put Adalberon in a predicament: having been elevated by Otto I to the powerful Archbishopric of Reims, he was forced to leave his episcopal seat and took refuge in one of his fortresses on the Meuse river, which belonged to the Ottonian sphere. The escape of the archbishop was perceived by Louis V as treason; he turned violently against Adalberon and threatened him with a siege of Reims. The matter was finally settled in a trial court at Compiègne. Before this meeting, however, Louis V changed his mind and sought a reconciliation with Adalberon, and so in the spring of 987, he planned a peace meeting with Empress Theophanu, who acted on behalf of her son Otto III. Before all these tangled events were resolved, Louis V died on 22 May 987 from a fall while hunting in the Forest of Halatte near the town of Senlis, Oise. He was buried in the Abbey of Saint-Corneille in Compiègne.

Louis V left no legitimate heirs, so his uncle Charles, Duke of Lower Lorraine, was nominated as the hereditary successor to the throne. But the clergy, including both Adalberon and Gerbert (who later became Pope Sylvester II), argued eloquently for the election of Hugh Capet, who was not only of royal blood but had proven himself through his actions and his military might. Hugh was elected to the Frankish throne and Adalberon crowned him, all within two months of Louis V's death. Thus the rule of the Carolingian dynasty ended and the Capetian era had begun.

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

title: Louis V of France


text:
Talk:Guadeloupe

A Commons file used on this page or its Wikidata item has been nominated for deletion
The following Wikimedia Commons file used on this page or its Wikidata item has been nominated for deletion: Participate in the deletion discussion at the. —Community Tech bot (talk) 14:38, 22 May 2023 (UTC)
 * Panorama Chutes du Carbet.jpg

title: Talk:Guadeloupe


text:
Grandfather clause

A grandfather clause, also known as grandfather policy, grandfathering, or being grandfathered in, is a provision in which an old rule continues to apply to some existing situations while a new rule will apply to all future cases. Those exempt from the new rule are said to have grandfather rights or acquired rights, or to have been grandfathered in. Frequently, the exemption is limited, as it may extend for a set time, or it may be lost under certain circumstances; for example, a grandfathered power plant might be exempt from new, more restrictive pollution laws, but the exception may be revoked and the new rules would apply if the plant were expanded. Often, such a provision is used as a compromise or out of practicality, to allow new rules to be enacted without upsetting a well-established logistical or political situation. This extends the idea of a rule not being retroactively applied.

Southern United States
The term originated in late 19th-century legislation and constitutional amendments passed by a number of Southern U.S. states, which created new requirements for literacy tests, payment of poll taxes and residency and property restrictions to register to vote. States in some cases exempted those whose ancestors (e.g., grandfathers) had the right to vote before the American Civil War or as of a particular date from such requirements. The intent and effect of such rules was to prevent former African-American slaves and their descendants from voting but without denying poor and illiterate whites the right to vote. Although these original grandfather clauses were eventually ruled unconstitutional, the terms grandfather clause and grandfather have been adapted to other uses.

The original grandfather clauses were contained in new state constitutions and Jim Crow laws passed between 1890 and 1908 by white-dominated state legislatures including Alabama, Georgia, Louisiana, North Carolina, Oklahoma, and Virginia. They restricted voter registration, effectively preventing African Americans from voting. Racial restrictions on voting in place before 1870 were invalidated by the Fifteenth Amendment.

After Democrats took control of state legislatures again, especially after the Compromise of 1877, they began to work to restrict the ability of black people to vote. Paramilitary groups such as the White League, Red Shirts, and rifle clubs had intimidated black people or barred them from the polls in numerous elections before what they called the Redemption (restoration of white supremacy). Nonetheless, a coalition of Populists and Republicans in fusion tickets in the 1880s and 1890s gained some seats and won some governor positions. To prevent such coalitions in the future, the Democrats wanted to exclude freedmen and other black people from voting; in some states they also restricted poor whites to avoid biracial coalitions.

White Democrats developed statutes and passed new constitutions creating restrictive voter registration rules. Examples included imposition of poll taxes and residency and literacy tests. An exemption to such requirements was made for all persons allowed to vote before the American Civil War, and any of their descendants. The term grandfather clause arose from the fact that the laws tied the then-current generation's voting rights to those of their grandfathers. According to Black's Law Dictionary, some Southern states adopted constitutional provisions exempting from the literacy requirements descendants of those who fought in the army or navy of the United States or of the Confederate States during a time of war.

After the U.S. Supreme Court found such provisions unconstitutional in Guinn v. United States (1915), states were forced to stop using the grandfather clauses to provide exemption to literacy tests. Without the grandfather clauses, tens of thousands of poor Southern whites were disenfranchised in the early 20th century. As decades passed, Southern states tended to expand the franchise for poor whites, but most black people could not vote until after passage of the 1965 Voting Rights Act. Ratification in 1964 of the Twenty-fourth Amendment to the United States Constitution prohibited the use of poll taxes in federal elections, but some states continued to use them in state elections.

The 1965 Voting Rights Act had provisions to protect voter registration and access to elections, with federal enforcement and supervision where necessary. In 1966, the Supreme Court ruled in Harper v. Virginia Board of Elections that poll taxes could not be used in any elections. This secured the franchise for most citizens, and voter registration and turnout climbed dramatically in Southern states.

Other contexts
There is also a rather different, older type of grandfather clause, perhaps more properly a grandfather principle in which a government blots out transactions of the recent past, usually those of a predecessor government. The modern analogue may be repudiating public debt, but the original was Henry II's principle, preserved in many of his judgments, "Let it be as it was on the day of my grandfather's death", a principle by which he repudiated all the royal grants that had been made in the previous 19 years under King Stephen.

title: Grandfather clause


text:
Painted Angels

Painted Angels (1997) (US: The Wicked Wicked West) is a film by Jon Sanders starring Brenda Fricker, Kelly McGillis, Meret Becker, Bronagh Gallagher, Lisa Jakub and Anna Mottram.

The film was shot in Saskatchewan, Canada, and follows the lives of several women in a brothel in a midwestern prairie town in the 1870s. It premièred at the International Film Festival Rotterdam and was released in cinemas in the UK by Artificial Eye in February 1999 and on video in June 1999.

title: Painted Angels


text:
Amoghavajra

Amoghavajra (अमोघवज्र ;, 705–774) was a prolific translator who became one of the most politically powerful Buddhist monks in Chinese history and is acknowledged as one of the Eight Patriarchs of the Doctrine in Shingon Buddhism.

Life
Born in Samarkand of an Indian merchant or Brahmin father as according to Feixis's Stele Inscription and a mother of Sogdian origin, he went to China at age 10 after his father's death. In 719, he was ordained into the sangha by Vajrabodhi and became his disciple. After all foreign monks were expelled from China in 741, he and some associates went on a pilgrimage to gather texts, visiting Sri Lanka, Indochina and India. During this voyage, he apparently met Nagabodhi, Vajrabodhi's master, and studied the Tattvasaṃgraha Tantra at length. He returned to China in 746 with some five hundred volumes gathered from Sri Lanka.

In 750, he left the court to join the military governorship of General Geshu Han of the Tang dynasty, for whom he conducted large-scale tantric initiations at his field headquarters. In 754, he translated the first portion of the Tattvasaṃgraha Tantra texts (Taishō Tripiṭaka. 865), central to the Outer Tantras of Vajrayana Buddhism, which became one of his most significant accomplishments. He regarded its teachings as the most effective method for attaining enlightenment yet devised, and incorporated its basic schema in a number of writings.

Amoghavajra was captured during the An Lushan Rebellion but in 757 was freed by loyalist forces, whereupon he performed rites to purify the capital and consolidate the security of the Tang state. Two years later, he initiated the emperor Emperor Suzong of Tang as a chakravartin.

Amoghavajra assisted the Tang dynasty state against the An Lushan rebellion. He carried out Vajrayana rituals which were ostensibly effective in supernaturally attacking and destroying An Lushan's army including the death of one of An Lushan's generals, Zhou Zhiguang.

In 765, Amoghavajra used his new rendition of the Humane King Sutra in an elaborate ritual to counter the advance of a 200,000-strong army of Tibetan and Uyghurs which was poised to invade Chang'an. Its leader, Pugu Huai'en, dropped dead in camp and his forces dispersed.

The opulent Jinge Temple on Mount Wutai was completed in 767, a pet project of Amoghavajra, one of his many efforts to promote the bodhisattva Mañjuśrī as the protector of China. Amoghavajra continued to perform rites to avert disaster at the request of Emperor Daizong of Tang.

On his death in 774, three days of mourning were officially declared, and he posthumously received various exalted titles. The Chinese monks Huilang, Huiguo and Huilin were among his most prominent successors. Seventy-seven texts were translated by Amoghavajra according to his own account, though many more, including original compositions, are ascribed to him in the Chinese canons.

title: Amoghavajra


text:
User talk:Fjkraan

Main interest: Micro computer history.

Site: Information on computers in my collection

title: User talk:Fjkraan


text:
Aleph number

In mathematics, particularly in set theory, the aleph numbers are a sequence of numbers used to represent the cardinality (or size) of infinite sets that can be well-ordered. They were introduced by the mathematician Georg Cantor and are named after the symbol he used to denote them, the Hebrew letter aleph (ℵ).

The cardinality of the natural numbers is ℵ0 (read aleph-nought or aleph-zero or aleph-null), the next larger cardinality of a well-ordered set is aleph-one ℵ1, then ℵ2 and so on. Continuing in this manner, it is possible to define a cardinal number ℵα for every ordinal number α, as described below.

The concept and notation are due to Georg Cantor, who defined the notion of cardinality and realized that infinite sets can have different cardinalities.

The aleph numbers differ from the infinity (∞) commonly found in algebra and calculus, in that the alephs measure the sizes of sets, while infinity is commonly defined either as an extreme limit of the real number line (applied to a function or sequence that "diverges to infinity" or "increases without bound"), or as an extreme point of the extended real number line.

Aleph-zero
ℵ0 (aleph-zero, also aleph-nought or aleph-null) is the cardinality of the set of all natural numbers, and is an infinite cardinal. The set of all finite ordinals, called ω or ω0 (where ω is the lowercase Greek letter omega), has cardinality ℵ0. A set has cardinality ℵ0 if and only if it is countably infinite, that is, there is a bijection (one-to-one correspondence) between it and the natural numbers. Examples of such sets are

 * the set of natural numbers, irrespective of including or excluding zero,
 * the set of all integers,
 * any infinite subset of the integers, such as the set of all square numbers or the set of all prime numbers,
 * the set of all rational numbers,
 * the set of all constructible numbers (in the geometric sense),
 * the set of all algebraic numbers,
 * the set of all computable numbers,
 * the set of all computable functions,
 * the set of all binary strings of finite length, and
 * the set of all finite subsets of any given countably infinite set.

These infinite ordinals: ω, ω + 1, ω⋅2, ω2, ωω, and ε0 are among the countably infinite sets. For example, the sequence (with ordinality ω⋅2) of all positive odd integers followed by all positive even integers

 * {1, 3, 5, 7, 9, ...; 2, 4, 6, 8, 10, ...}

is an ordering of the set (with cardinality ℵ0) of positive integers.

If the axiom of countable choice (a weaker version of the axiom of choice) holds, then ℵ0 is smaller than any other infinite cardinal, and is therefore the (unique) least infinite ordinal.

Aleph-one
ℵ1 is, by definition, the cardinality of the set of all countable ordinal numbers. This set is denoted by ω1 (or sometimes Ω). The set ω1 is itself an ordinal number larger than all countable ones, so it is an uncountable set. Therefore, ℵ1 is distinct from ℵ0. The definition of ℵ1 implies (in ZF, Zermelo–Fraenkel set theory without the axiom of choice) that no cardinal number is between ℵ0 and ℵ1. If the axiom of choice is used, it can be further proved that the class of cardinal numbers is totally ordered, and thus ℵ1 is the second-smallest infinite cardinal number. One can show one of the most useful properties of the set ω1: Any countable subset of ω1 has an upper bound in ω1 (this follows from the fact that the union of a countable number of countable sets is itself countable). This fact is analogous to the situation in ℵ0: Every finite set of natural numbers has a maximum which is also a natural number, and finite unions of finite sets are finite.

The ordinal ω1 is actually a useful concept, if somewhat exotic-sounding. An example application is "closing" with respect to countable operations; e.g., trying to explicitly describe the σ-algebra generated by an arbitrary collection of subsets (see e.g. Borel hierarchy). This is harder than most explicit descriptions of "generation" in algebra (vector spaces, groups, etc.) because in those cases we only have to close with respect to finite operations – sums, products, etc. The process involves defining, for each countable ordinal, via transfinite induction, a set by "throwing in" all possible countable unions and complements, and taking the union of all that over all of ω1.

Continuum hypothesis
The cardinality of the set of real numbers (cardinality of the continuum) is 2ℵ0. It cannot be determined from ZFC (Zermelo–Fraenkel set theory augmented with the axiom of choice) where this number fits exactly in the aleph number hierarchy, but it follows from ZFC that the continuum hypothesis (CH) is equivalent to the identity
 * 2ℵ0 = ℵ1.

The CH states that there is no set whose cardinality is strictly between that of the integers and the real numbers. CH is independent of ZFC: It can be neither proven nor disproven within the context of that axiom system (provided that ZFC is consistent). That CH is consistent with ZFC was demonstrated by Kurt Gödel in 1940, when he showed that its negation is not a theorem of ZFC. That it is independent of ZFC was demonstrated by Paul Cohen in 1963, when he showed conversely that the CH itself is not a theorem of ZFC – by the (then-novel) method of forcing.

Aleph-omega
Aleph-omega is
 * ℵω = sup{ ℵn | n ∈ ω } = sup{ ℵn | n ∈ {0, 1, 2, ...} }

where the smallest infinite ordinal is denoted as ω. That is, the cardinal number ℵω is the least upper bound of
 * { ℵn | n ∈ {0, 1, 2, ...} }.

Notably, ℵω is the first uncountable cardinal number that can be demonstrated within Zermelo–Fraenkel set theory not to be equal to the cardinality of the set of all real numbers 2ℵ0: For any natural number n ≥ 1, we can consistently assume that 2ℵ0 = ℵn, and moreover it is possible to assume that 2ℵ0 is as least as large as any cardinal number we like. The main restriction ZFC puts on the value of 2ℵ0 is that it cannot equal certain special cardinals with cofinality ℵ0. An uncountably infinite cardinal κ having cofinality ℵ0 means that there is a (countable-length) sequence κ0 ≤ κ1 ≤ κ2 ≤ ... of cardinals κi < κ whose limit (i.e. its least upper bound) is κ (see Easton's theorem). As per the definition above, ℵω is the limit of a countable-length sequence of smaller cardinals.

Aleph-α for general α
To define ℵα for arbitrary ordinal number α, we must define the successor cardinal operation, which assigns to any cardinal number ρ the next larger well-ordered cardinal ρ+ (if the axiom of choice holds, this is the (unique) next larger cardinal).

We can then define the aleph numbers as follows:

 * ℵ0 = ω
 * ℵα+1 = (ℵα)+
 * ℵλ = ⋃{ ℵα | α < λ } for λ an infinite limit ordinal,

The α-th infinite initial ordinal is written ωα. Its cardinality is written ℵα.

Informally, the aleph function ℵ: On → Cd is a bijection from the ordinals to the infinite cardinals. Formally, in ZFC, ℵ is not a function, but a function-like class, as it is not a set (due to the Burali-Forti paradox).

Fixed points of omega
For any ordinal α we have
 * α ≤ ωα.

In many cases ωα is strictly greater than α. For example, it is true for any successor ordinal: α + 1 < ωα+1 holds. There are, however, some limit ordinals which are fixed points of the omega function, because of the fixed-point lemma for normal functions. The first such is the limit of the sequence

 * ω, ωω, ωω ω, ...,

which is sometimes denoted ωω ... .

Any weakly inaccessible cardinal is also a fixed point of the aleph function. This can be shown in ZFC as follows. Suppose κ = ℵλ is a weakly inaccessible cardinal. If λ were a successor ordinal, then ℵλ would be a successor cardinal and hence not weakly inaccessible. If λ were a limit ordinal less than κ then its cofinality (and thus the cofinality of ℵλ) would be less than κ and so κ would not be regular and thus not weakly inaccessible. Thus λ ≥ κ and consequently λ = κ which makes it a fixed point.

Role of axiom of choice
The cardinality of any infinite ordinal number is an aleph number. Every aleph is the cardinality of some ordinal. The least of these is its initial ordinal. Any set whose cardinality is an aleph is equinumerous with an ordinal and is thus well-orderable.

Each finite set is well-orderable, but does not have an aleph as its cardinality.

Over ZF, the assumption that the cardinality of each infinite set is an aleph number is equivalent to the existence of a well-ordering of every set, which in turn is equivalent to the axiom of choice. ZFC set theory, which includes the axiom of choice, implies that every infinite set has an aleph number as its cardinality (i.e. is equinumerous with its initial ordinal), and thus the initial ordinals of the aleph numbers serve as a class of representatives for all possible infinite cardinal numbers.

When cardinality is studied in ZF without the axiom of choice, it is no longer possible to prove that each infinite set has some aleph number as its cardinality; the sets whose cardinality is an aleph number are exactly the infinite sets that can be well-ordered. The method of Scott's trick is sometimes used as an alternative way to construct representatives for cardinal numbers in the setting of ZF. For example, one can define card( S ) to be the set of sets with the same cardinality as S of minimum possible rank. This has the property that card( S ) = card( T ) if and only if S and T have the same cardinality. (The set card( S ) does not have the same cardinality of S in general, but all its elements do.)

title: Aleph number


text:
Pico Island

Pico Island (Ilha do Pico, ) is an island in the Central Group of the Portuguese Azores. The landscape features an eponymous volcano, Ponta do Pico, which is the highest mountain in Portugal, the Azores, and the highest elevation of the Mid-Atlantic Ridge. In the tradition of the Portuguese poet, Raul Brandão, Pico is referred to as the Ilha Preta ("Black Island"), for its black volcanic soils, which nourish its UNESCO-designated vineyards that once allowed the development of the island's economy. Pico is the second largest and, geologically speaking, the most recently formed island of the Azores, being around 300,000 years old.

History

The exact date of the island's discovery is not known. However, in the 1375 Catalan Atlas, Pico is depicted along several other islands of the Azores, where it is labelled as li colunbj. In Cristoforo Soligo's map from 1475, Pico is described both as Dom Diniz (or São Diniz) island and as Insula Columbi.

Although the Azores were inhabited since 1439, settlement on Pico only started in the 1480s when Pico became part of the neighboring island of Faial's control. Pico is thought to be the last of the islands of the Central Group to be occupied.

In a letter dated March 28, 1481, the administrator of the Azores, D. Beatriz, gave the possession of the island of Pico to the first captain of the donatarie, D. Álvaro d'Ornelas, a resident of island of Madeira, on the condition that he populate the island ('Captains', often of noble heritage, obtained land rights in regions not important enough to merit a 'governor', and were a cheap way for the state to settle new territory). d'Ornelas failed to attract settlers, so on December 29, 1482, the Flemish captain of the island of Faial, 7 km from Pico, Jós d'Hutra, became the new captain. The transfer of the island of Pico to Jós d'Hutra not only confirms that Pico was not colonized at the end of the 15th century, but also demonstrates the interest in boosting the insular population of the islands of the central group. This was understandable given the growing geo-economic importance of the Azores archipelago at that time, as the Portuguese monarchy was expanding into Morocco and along the west African coast.

Population grew slowly, in part because of difficulties cultivating wheat. By 1587, there were less than 3,000 inhabitants. Soil quality is conditioned by the volcanic soils consisting mostly of lightly weathered recent basalt (designated as "biscoitos" or "mistérios"). Examples include the Mistério de São João, resulting from a 1718 eruption. In addition, there are few rivers on the island compared to other parts of the archipelago. The last eruptions of magma took place in 1718 and 1720 and generated three volcanic cinder cones (Mistérios): Santa Luzia and São João in 1718 and Silveira in 1720.

Aside from a few slaves sent to herd sheep in isolation before European settlement, the island's first inhabitants were primarily Portuguese, but of diverse origins. Settlers of Portuguese origin arrived from different parts of the metropolis, some passing though Madeira Island or other Azorean islands, particularly Terceira. Some individuals had noble origins, owning land and goods; there were secular and regular clergy, especially the Franciscans; and another group consisted of merchants, craftsmen, farmers and artisans. A small number of Jews came from the beginning of the 16th century, and there were indentured slaves of African origin at this time who farmed and did domestic work. The first municipality in Pico was established at Lajes, where fresh water was available (the place-name (Ribeiras) proves this).

Around 9,000 lived on Pico by 1695. Pico moved "up the ladder" of urban hierarchies from fifth largest centre in the Azores to fourth, behind São Miguel, Terceira and Faial. By the middle of the 18th century the population was over 19,000 people. In 1542, the inhabitants of the north of the island asked King João III to create a second village, to allow better law and order in a frontier economy with theft and corruption. São Roque do Pico was established. It was only much later, when the link between the islands of Pico and Faial strengthened further, that there was a need to create a third municipality. Madalena was established in 1723 (the larger town of Horta is 7 km away on Faial, and was where many of the island's property-owners and winemakers lived).

The islands of Pico and Faial were administered jointly until 1766, when Pico became an integral part of the Province of the Azores, led by a captain-general based in Angra. Centralizing governance in this way had poor success: in 1836, the people of Pico were integrated into a smaller administrative unit again, the third Azorean district, with the islands of Faial, Flores and Corvo.

Settlers initially farmed wheat and some olives in addition to developing a woad industry, as on Faial (based on lichens that were exported to Flanders to produce commercial dyes). The two islands often shared livelihood activities Viticulture, helped by the richer soils found in suitable micro-climates, then grew rapidly. The industry was disturbed by the last volcanic eruptions, but viticulture expanded commercial livelihood opportunities considerably. Pico's famous verdelho wine production, dating back two centuries, was widely appreciated in many countries (including England and in the Americas and even in the palaces of the Russian czars). Viticulture declined with the spread of powdery mildew and phylloxera during the middle of the 19th century, creating a crisis on the island that lasted until the 20th century.

The other major economic activity was whaling. The presence of American and British whalers in the waters of the Azores at the end of the 18th century stabilized the economy. Spermwhale hunting became the primary industry around the island until it was abolished in 1986. The whale factory is now a museum.

Physical geography
The island is 17.5 km south of São Jorge and just 7 km east of Faial, in the Central Group of islands, an area that  is colloquially known as O Triângulo (The Triangle). Pico is 46.3 kilometers long, and at its maximum extent about 16 kilometers wide; making it the second largest of the Azores islands. The 151.8 km coastline is slightly indented with inlets and coves. Accessibility to the sea poses no difficulties, except in sections with plunging, unstable cliffs, subject to frequent landslides (as seen in the northeast and southeast). Seabirds find favorable conditions for nesting in these habitats, in particular the species of Tern (Sterna hirundo and Sterna dougallii) and Shearwater (Calonectris borealis). The rocky substrates prevail throughout the coast (slabs that alternate with rolled pebbles), there are no sandy beaches, but there are numerous tide pools and natural inlets used for bathing.

Along the central plain of volcano cones to the eastern coast, the landscape is pitted with the remnants of craters and lake-filled cones, and dominated by the ancient volcano of Topo. Geomorphologically, the island is composed of three units:
 * Topo Volcano - Located on the southernmost side of the island, it corresponds to an ancient shield volcano created by the overlap of various lava flows and destroyed by erosion and landslides;
 * Achada Plateau - Extending from the east side of the island to the stratovolcano, the Achada Plateau is an axial zone comprising spatter and lava cones along a mountainous ridge filled with lake-filled craters, dense scrub and forests. Cones in this region are oriented along a west-northwest and south-southeast axis along a path east to west between the other volcanic structures;
 * Pico Volcano - It occupies the western portion of the island, and corresponds to a central volcano with 2351 m in altitude, created by numerous Hawaiian and Strombolian eruptions. Along its flanks are diverse lava and spatter cones that are formed along radial fractures and faults.

Generally, Pico Island's tectonic structures are oriented along a west-northwest to east-southeast and a northeast to southwest axis. The main axis controls the main structures, especially the main mountain of Pico, while the secondary axises affect the radial fractures and faults along the central plain and eastern volcano.

Similar to the geomorphological structures described above, the volcanic landforms have been identified by Madeira (1998) in the following units:

 * Volcanic Complex of Lajes - pertains to the oldest volcanic structures, comprising volcanoes, lavas and debris that are between 35 and 300,000 thousand years old; it makes up the structures of the eastern part of the island, including the ancient volcano of Topo.
 * Volcanic Complex of Calheta de Nesquim - dating to about 230,000 thousand years ago, this is a fissural zone of cones and basaltic lavas, associated with the Achada Plain, and the many faults along the central plateau of the island.
 * Volcanic Complex of Madalena - the most recent volcanic complex, with both geological and historical eruptions; it includes the basaltic stratovolcano of Pico, but also many of the cones, faults and lava fields that circle the 2,351 meter summit.

The last volcanic eruption (which is debatable) occurred in 1963, in a small submarine eruption off the northwest coast (north of Cachorro in Santa Luzia). Prior to this, major volcanic activity was evident in lava fields generated in only the areas of Prainha (1562–64), Santa Luzia (1718), São João (1718) and Silveira (1720). The paths of the lava flows are still visible; those in the 16th century and 1718 were particularly substantial, extending for over 10 km. Today, the only visible evidence of active volcanism appears on the summit of Pico, in the base and interior of the Piqueninho (the mini-peak on the summit); there are irregular emissions from cracks in this zone. Additionally, there are emissions on the eastern flanks between 1500 and 2000 meters altitude. There have also been found areas of de-gasification along the fault associated with the Lagoa do Capitão and another in the Topo Volcano, as well as a spring rich in in the area of Silveira.

Earthquakes
Pico, much like the other islands, is susceptible to seismic events, although their epicenters have primarily been localized in the Faial-Pico or Pico-São Jorge Channels. Only three significant tectonic earthquakes have hit the island of Pico since the beginning of settlement: The 1757 earthquake, which caused significant damage in the eastern region of the island including the death of 11 people and the collapse of a church and a few houses; the 1973 seismic crisis (October 1973 - May 1974) in which the main earthquake (with magnitude 5.5) did considerable damage to 604 houses in Faial and 2,000 in Pico, which left nearly 5,000 people homeless; and the July 9, 1998 earthquake, the strongest earthquake registered in the last 30 years, which had its epicenter 5 kilometers northeast of Ponta da Ribeirinha, on the island of Faial, reaching a 5.8 magnitude. It was felt on Pico, with a maximum intensity level of VII on the Mercalli scale; some homes were damaged and possessions were buffeted. Other violent earthquakes which have affected the island include: the 1957-58 seismic events associated with the eruption of the Capelinhos volcano (on Faial), the February 1964 earthquake on the island of São Jorge, and the 1926 earthquake, whose epicenter was registered in the Faial-Pico Channel.

Biome

According to the 2018 land occupation chart of the Azores, compared to the other islands of the archipelago, Pico has, both by proportion and total land area, the largest area of intact native laurisilva forests, and broadleaf forests (which includes introduced species such as Pittosporum undulatum), that occupy, respectively, 2.9% and 30% of the island. Pico is the only island in the Azores where pasture and meadow are not the dominant landscape (accounting only for around 29% of the territory).

Like the remaining islands of the Azores, Pico suffered extensive deforestation following its occupation. While some land, especially at the coast, was used for housing, the majority was and is still used for farming, livestock and the wood industry (of Cryptomeria). Additionally, several introductions of exotic species in the 19th century, like Hydrangea macrophylla, Pittosporum undulatum, Hedychium gardnerianum and Psidium cattleyanum, resulted in invasion and destruction of the remaining native habitats.

Gaspar Frutuoso gave a brief description of the island's flora only a century after being occupied:

"'All around the island there is plenty and large wood of cedar [Juniperus brevifolia], sanguinho [Frangula azorica], sour cherry [Prunus lusitanica], pau branco [Picconia azorica], faias [Morella faya], laurels [Laurus azorica], and above all, the yew wood [Taxus baccata], found solely on Pico [...] It has many groves: Cedar, pau branco, laurel, faia, tamujo [Myrsine africana], heaths [Erica azorica] as large as trees, sanguinhos, zimbro and laurustinus [Viburnum treleasei] [...]'"

Over a third of the Natura 2000 protected areas of the Azores are located on Pico. It has also the largest number of volcanic cavities (lava caves) of anywhere in the archipelago, 86 out of 225.

Climate
Pico, like the remaining islands of the archipelago, has a mild maritime climate with high precipitation for most of the year, high humidity and constant winds. However, due to its unusual topographic variation, temperature and precipitation levels vary significantly depending on both altitude and location (generally, rainfall is more concentrated in the east). Temperatures vary around 8 °C between the coolest and the warmest month (February and August respectively)

At lower elevations, where most of the population resides, Pico has a humid subtropical climate (Köppen: Cfa; Thornthwaite: B) with mild, rainy winters (mean around 14.5 C) and warm, noticeably drier summers, (mean around 22 C) although generally not dry enough to be classified as Mediterranean. It has an average annual temperature of 17.5 - 18 C and around 1000 to 1600 mm of precipitation fall each year, though values as high as 2000 mm are typical on the central northeast coast (São Roque, Prainha and Santo Amaro). Because of its high relative humidity (around 80%), summers can feel slightly warmer than they really are.

At higher elevations, like the Achada Plateau, temperatures are cooler and precipitation increases, it therefore transitions into a wet oceanic climate (Köppen: Cfb; Thornthwaite: A), average relative humidity stays around 90% and fogs produced by the cooling of rising air masses from the ocean are persistent, which, in turn, creates an environment typical of a cloud forest. Places such as the Caiado and Paul lakes receive, respectively, 4698 and 3357 mm of precipitation each year and Mount Pico is recognized as receiving the largest annual rainfall in Portugal (over 5000 mm per year at the northeastern flanks of the volcano).

At around 1200 m, the climate becomes subpolar oceanic (Cfc) (somewhat comparable to the Faroe Islands). As altitude surpasses 1500 m and exceeds the planetary boundary layer, both precipitation and humidity levels decrease reaching the top of Mount Pico which averages only around 500 mm of precipitation. Temperatures at the peak of the stratovolcano (~2300 m a.s.l.) average 2 C and snow can be frequently seen in the cooler months, one of the few places in the archipelago where this phenomenon is possible. The climate is classified as alpine at the very top.

Pico has registered the highest temperature ever recorded in the Azores, 32.1 C on 7 September 1985 at Pico Airport and although there are no climatic stations at Mount Pico, it is likely that the island has also experienced the lowest temperatures in the archipelago at the top of the mountain.

Human geography
Geological, geomorphological and climatic constraints played (and still have) a determining role in economic activity, in the distribution of urban agglomerations and in demographic evolution. The settlement pattern is a good example of the native's adaptation to the specific physical geography of this island, developing mainly along the coast and along the roads of communication. The main urban centers, municipality seats, are located by the sea: Madalena, São Roque do Pico and Lajes do Pico.

Regarding demographic behavior, the population has continued to decline since the middle of the last century, a trend that continues to the present day. However, the 1960s and 1970s were the decades that showed a more negative population variation rate, due to the intense emigration surge that was generated at this time and that was generalized to the entire archipelago. Unlike other islands in the Azores, in the 1990s the situation did not change, continuing to register a negative population variation (-2.6%), even higher than that registered in the 1980s. According to the XIV General Census of Population (INE, 2002), the population totaled 14,806 inhabitants, which represents a decrease of 396 inhabitants compared to 1991. In 2001, the municipality of Madalena had the largest population (6,136 individuals), followed by Lajes do Pico (5,041 individuals) and São Roque (3,629 individuals). The population density (33.3 inhabitants/km2) was among the lowest in the archipelago and the age structure showed signs of aging, with more than 50% of the population aged between 25 and 64 years, against 30% of young people and about 20% of seniors.

As of 2021, the total population is estimated to be around 13,895 inhabitants.

Economy
Historically, Pico was largely dependent on Faial for its economic, political and social activities, and it still is to this day, although mitigated by the entrepreneurial dynamics of its population and by the centralities created by the autonomous regime. The close relations with the neighboring island are still established in the maritime routes that cross the channel daily. Fishing is one of the main activities of the primary sector, namely the capture of tuna for the canning industry, which mainly employs female labor in the processing of fish. Dairy farms supply raw material to the various cheese factories, whose product is highly appreciated for its artisanal characteristics (soft paste).

Currently, it is the secondary and tertiary activity sectors that have the greatest dynamism, occupying around 30% and 50% of the active population, respectively, due to the growth in the offer of services and the installation of branches of the regional administration, which generate specialized employment. The employment rate increased in the 1990s, from 35% to 40%, and contrary to the regional average, the unemployment rate remained stable from 1991 to 2001 (3.2%). However, the level of education is low, with most inhabitants having only the 1st cycle of basic education. Note that only 4.3% have higher education, which is lower than the regional average (5.2%). This situation, together with the still significant percentage of working people in the primary sector (20%), indicates an unfavorable panorama in terms of professional qualifications among the active population.

Like the other islands, tourism is also an emerging activity in Pico, seen as a strategic pillar for diversifying the economic base. The offer of accommodation has progressed, seeking to respond to the needs of an increasingly demanding destination, with five hotel units (hotels, pensions and others) and four for tourism in rural areas (2002). The support services to visitors and the establishments of restaurants, culture, car rental, tourist entertainment, among others, multiply. As the main whaling center in the Azores, economic agents knew how to transform whale watching into a flourishing business, through the reactivation of an ancestral activity with an undeniable heritage of historical and cultural interest (boats, whaling boats, lookouts, former manufacturing industries, etc.). The capture of sperm whales (Physeter macrocephalus) for oil extraction and production of meal for animal feed definitively ceased in the 1980s, due to the replacement of whale-derived products and compliance with international conventions. The first operator of «whale watching» was installed in Lajes do Pico, with around 26 companies linked to nautical activities.

Pico Wine and vine culture

The vine culture on Pico Island began in the late 15th century, when the settlement of the island started. The Verdelho vines found ideal conditions to grow and mature in the volcanic soil and dark stone walls (currais). Later, it was exported to many countries in Europe and the Americas, and even reached the Russian court tables. Years later, wine production is not as important of an economic activity as it used to be, despite trials to improve the varieties and incentives for the restoration of «currais». The vineyards, which mark the island's landscape and ancestral cultural practices, are a UNESCO World Heritage Site since 2004 (see Landscape of the Pico Island Vineyard Culture).

Sustainable tourism
In 2012, Pico received the QualityCoast Gold Award for its efforts to become a sustainable tourism destination. Because of this award, Pico was selected for inclusion in the global atlas for sustainable tourism DestiNet.

Notable people

 * José da Costa Nunes (1880 in Candelaria do Pico – 1976) a Cardinal of the Roman Catholic Church, Bishop of Macau, 1920-1940 and Patriarch of the East Indies, 1940–1953.
 * José Vieira Alvernaz (1898 in Ribeirinha - 1986) a Portuguese prelate, Bishop of Cochin, Archbishop of Goa and Daman and Patriarch of the East Indies
 * Arquimínio Rodrigues da Costa (1924 in São Mateus – 2016) a prelate of the Catholic Church, Bishop of the Diocese of Macau, 1976–1988.
 * Fernando Machado Soares (1930 in São Roque do Pico – 2014) a fado singer, poet, composer, jurist and retired judge.
 * Manuel da Silva Rosa (born 1961 in Madalena) a Portuguese American writer about Christopher Columbus
 * Caesar DePaço (born 1965 in Madalena) a controversial businessman and former consul for both Portugal and Cape Verde
 * Duarte Freitas (born 1966 in São Roque do Pico) a politician and former Member of the European Parliament

title: Pico Island


text:
Talk:Thomas Pynchon

External links modified
Hello fellow Wikipedians,

I have just added archive links to 1 one external link on Thomas Pynchon. Please take a moment to review my edit. If necessary, add after the link to keep me from modifying it. Alternatively, you can add to keep me off the page altogether. I made the following changes:
 * Added archive http://web.archive.org/web/20061112151827/http://themodernword.com/pynchon/paper_gibbs.html to http://www.themodernword.com/pynchon/paper_gibbs.html

When you have finished reviewing my changes, please set the checked parameter below to true or failed to let others know (documentation at ).

Cheers.—cyberbot II  Talk to my owner :Online 17:00, 6 March 2016 (UTC)

External links modified
Hello fellow Wikipedians,

I have just modified 12 external links on Thomas Pynchon. Please take a moment to review my edit. If you have any questions, or need the bot to ignore the links, or the page altogether, please visit this simple FaQ for additional information. I made the following changes:
 * Added archive https://web.archive.org/web/20130315194714/http://themodernword.com/pynchon/pynchon_hamster.html to http://themodernword.com/pynchon/pynchon_hamster.html
 * Added archive https://web.archive.org/web/20130315194955/http://themodernword.com/pynchon/pynchon_boys.html to http://themodernword.com/pynchon/pynchon_boys.html
 * Added archive https://web.archive.org/web/20130119074615/http://themodernword.com/pynchon/pynchon_knight.html to http://themodernword.com/pynchon/pynchon_knight.html
 * Added archive https://web.archive.org/web/20090719002805/http://revel.unice.fr/cycnos/document.html?id=1475 to http://revel.unice.fr/cycnos/document.html?id=1475
 * Added archive https://web.archive.org/web/20100915171907/http://www.themodernword.com/pynchon/pynchon_grintro.html to http://www.themodernword.com/pynchon/pynchon_grintro.html
 * Added archive https://web.archive.org/web/20141015150615/http://www.themodernword.com/pynchon/review_nyt_vineland.html to http://themodernword.com/pynchon/review_nyt_vineland.html
 * Added archive https://web.archive.org/web/20101130115328/http://themodernword.com/pynchon/pynchon_film_tv.html to http://themodernword.com/pynchon/pynchon_film_tv.html
 * Added archive https://web.archive.org/web/20110311101734/http://www.themodernword.com/pynchon/ketzan_simpsons.htm to http://themodernword.com/pynchon/ketzan_simpsons.htm
 * Added archive https://web.archive.org/web/20101130114945/http://themodernword.com/pynchon/pynchon_essays_dailyshow.html to http://www.themodernword.com/pynchon/pynchon_essays_dailyshow.html
 * Added archive https://web.archive.org/web/20130603130333/https://www.pynchon.net/owap to https://www.pynchon.net/owap
 * Added archive https://web.archive.org/web/20071130094801/https://artfilm.ch/thomaspynchon.php to http://artfilm.ch/thomaspynchon.php
 * Added archive https://web.archive.org/web/20110629141909/https://isthistomorrow.com/archive/pynchon.html to http://isthistomorrow.com/archive/pynchon.html

When you have finished reviewing my changes, you may follow the instructions on the template below to fix any issues with the URLs.

Cheers.— InternetArchiveBot  (Report bug) 21:04, 20 May 2017 (UTC)

External links modified
Hello fellow Wikipedians,

I have just modified 2 external links on Thomas Pynchon. Please take a moment to review my edit. If you have any questions, or need the bot to ignore the links, or the page altogether, please visit this simple FaQ for additional information. I made the following changes:
 * Added archive https://web.archive.org/web/20060219074830/http://www.themodernword.com/pynchon/pynchon_essays_watts.html to http://www.themodernword.com/pynchon/pynchon_essays_watts.html
 * Added archive https://web.archive.org/web/20141203113826/http://cgi.cnn.com/US/9706/05/pynchon/ to http://cgi.cnn.com/US/9706/05/pynchon/

When you have finished reviewing my changes, you may follow the instructions on the template below to fix any issues with the URLs.

Cheers.— InternetArchiveBot  (Report bug) 11:54, 2 December 2017 (UTC)

Mary Tharaldsen Pynchon's First Wife?
Hello,

I've noticed as of late that Mary Tharaldsen has been listed as Pynchon's apparent first wife in the column of biographical information on the right of Pynchon's main Wikipedia page, with a date claiming that Pynchon and Tharaldsen had been married in 1967. This is the first time I've ever heard of this. While I know Pynchon and Tharaldsen were involved in the 60s, I have never come across any information stating they were married, at the time listed (again, 1967), or any other time for this matter. If this is in fact true, could someone please point me to the source cited for this? If no source can be provided, I believe the necessary course of action is obvious... — Preceding unsigned comment added by 2601:800:c003:6010:6c10:e98e:ba1e:6c00 (talk) 09:01, 29 August 2021 (UTC)

Reference to thomaspynchon.com
As far as I can see this is not his official presence online, so I think it should be moved to #See also ContributorOfKnowledge (talk) 11:38, 22 June 2023 (UTC)
 * You are right, but it should be moved to External links. Khiikiat (talk) 12:14, 22 June 2023 (UTC)

title: Talk:Thomas Pynchon


text:
Talk:Aleph number

incomplete
Need to define what the plus sign that looks like an exponent is in aleph sub alpha on the right of the equals sign in the section on aleph dash alpha for general alpha. 100.15.138.239 (talk) 23:27, 10 February 2017 (UTC)
 * It is defined in the previous paragraph. --Bill Cherowitzo (talk) 23:37, 10 February 2017 (UTC)

Implications of new research?
Just ran across this article this weekend: https://www.scientificamerican.com/article/mathematicians-measure-infinities-and-find-theyre-equal/ which seems to imply that all infinite sets have equal cardinality. It's been way to long since I studied any set theory for me to tie that into aleph numbers, but it's probably worth a mention. 38.109.137.2 (talk) 15:12, 18 September 2017 (UTC)
 * The title of that article is clumsily worded. No, it doesn't say at all that all infinite sets have the same cardinality.  It says that two particular infinite cardinalities, ones important enough to have been given names ($$\mathfrak{p}$$ and $$\mathfrak{t}$$), turn out to be provably equal, which was contrary to the previous expectation.
 * This isn't the right place to go into detail. I'd love to say more at the mathematics reference desk, if you'd care to ask a question there. --Trovatore (talk) 16:57, 18 September 2017 (UTC)
 * See Talk:Continuum hypothesis for a related discussion. JRSpriggs (talk) 05:25, 19 September 2017 (UTC)

The continuum as aleph_n
Maybe I am missing something essential, but it seems to me that the statement 'for any positive integer n we can consistently assume that $$2^{\aleph_0} = \aleph_n$$' cannot possibly be correct, because it would imply that all $$\aleph _{n}$$ are equivalent. — Preceding unsigned comment added by FRZH (talk • contribs) 01:49, 3 February 2020 (UTC)

 * Notice that that is "for any [one] positive integer", not "for ALL positive integers simultaneously". So while it implies that $$2^{\aleph_0} = \aleph_3$$ could be true or $$2^{\aleph_0} = \aleph_{25}$$ could be true, it does not imply that $$\aleph_3 = \aleph_{25}$$ is true. See Easton's theorem for more information. JRSpriggs (talk) 07:55, 3 February 2020 (UTC)

I think that I finally got your point, but still the formulation in the article is confusing, in particular the phrasing "we can consistently assume". If I understand correctly what you mean is that WITHIN ZF theory, it is not inconsistent to assume $$2^{\aleph_0} = \aleph_n$$, for any given n. — Preceding unsigned comment added by FRZH (talk • contribs) 22:48, 30 March 2020 (UTC)
 * I think you have a valid criticism. The precise formulation (of this limited part of the claim) is that, for any n, it is consistent with ZFC that $$2^{\aleph_0} = \aleph_n$$.  Also the "we" language is disfavored for reasons of encyclopedic tone; the sentence could be rewritten to avoid this.
 * The more general claim is a little harder to explain. We'd like to say something like, "for any ordinal α, with blah blah blah exceptions, it is consistent with ZFC that $$2^{\aleph_0}=\aleph_\alpha$$".
 * But we can't really say that, because it doesn't actually mean anything, at least taking "consistent" to mean "you can't prove a contradiction". That's because we can't talk about an arbitrary ordinal in the same language we use to talk about whether things can be proved.
 * The statement can be made precise by using a model-theoretic rather than proof-theoretic notion of consistency, but this requires special care to make sure the concept you come up with still reflects what you want. It's tricky to formulate in an elementary article.  --Trovatore (talk) 23:31, 30 March 2020 (UTC)

A union of countable sets is countable. Why do you need the axiom of choice to prove that?
I am writing about the following sentence in parentheses which appears in the main text of this article:

"(This follows from the fact that the union of a countable number of countable sets is itself countable, one of the most common applications of the axiom of choice.)"

Sorry, as a non-expert in these matters I cannot see why one needs the axiom of choice here.

For a non-expert like me it seems that the fact that

(*) "a union of countable number of DISJOINT countable sets is itself countable"

has a trivial proof, with no need for the axiom of choice. So, it seems to me that one way to deduce that

(**) "a union of countable number of (not necessarily disjoint) countable sets is itself countable"

would be to first show that if A and B are countable sets, then their difference A\B (the set of elements of A which is not in B) is also countable and then to apply (**).

Here again, I cannot see where the axiom of choice is needed. If it really is needed then it would be a good idea to explain why, for the benefit of non-experts like me. Thank you.

I suppose that my question is equivalent to the following question: Do you really need the axiom of choice to decide whether or not a given(an explicitly given?) element of A is also in B ? If so, why?

79.180.250.201 (talk) 15:51, 15 May 2020 (UTC)
 * No, your (*) is not provable without (a weak fragment of) the axiom of choice. You don't need much choice, but you need a little bit, to pick one enumeration for each of your pairwise disjoint countable sets.
 * If you want more details on that, please ask at WP:RD/Math, which is the right place to ask and which could use some interesting questions like that. As to whether we should include something about that in the article — not sure; don't have time to think about it right now, but maybe someone else should. --Trovatore (talk) 16:30, 15 May 2020 (UTC)

 * According to countable set, "a set S is countable if there exists an injective function f from S to the natural numbers". Suppose T is a countable set of disjoint countable sets. And suppose that $$S = \bigcup T$$. What is the function f which maps S injectively into ω? More specifically, if g is the function which maps T into ω, is 5 in the range of f and, if so, which element of S maps to 5? JRSpriggs (talk) 05:39, 16 May 2020 (UTC)

 * Gee, I dunno. Can't you just choose some member of S for each integer? 67.198.37.16 (talk) 03:16, 25 March 2024 (UTC)

Aleph-naught vs. aleph-nought
I think that "aleph-naught" is a misspelling of "aleph-nought". It is not just a matter of American vs. British spelling. Both "naught" and "nought" are words, but "naught" means "nothing", and only "nought" is used to signify the mathematical quantity 0, which is what is relevant here. If there is agreement on this, feel free to fix the article. Ebony Jackson (talk) 04:22, 12 November 2020 (UTC)
 * I've always understood subscript-zero to be pronounced as "naught" with an "a", not just for $$\aleph_0$$ but in general (say, for a variable called $$x_0$$). --Trovatore (talk) 04:27, 12 November 2020 (UTC)
 * I tried doing a MathSciNet search for published articles whose titles contain aleph-naught or aleph-nought. The search for aleph-naught turned up naught!  But there were several with aleph-nought:
 * Goodearl, K. R. Directly finite aleph-nought-continuous regular rings. Pacific J. Math. 100 (1982), no. 1, 105–122.
 * Ara, Pere Aleph-nought-continuous regular rings. J. Algebra 109 (1987), no. 1, 115–126.
 * Ara, Pere Stable range of aleph-nought-continuous regular rings. Ring theory (Granada, 1986), 1–7, Lecture Notes in Math., 1328, Springer, Berlin, 1988.
 * Wu, Tongsuo Exchange rings with general aleph-nought-comparability. Comm. Algebra 29 (2001), no. 2, 815–828.
 * Li, Qisheng; Tong, Wenting A remark on aleph-nought-comparability over exchange rings. Nanjing Daxue Xuebao Shuxue Bannian Kan 20 (2003), no. 1, 1–6.
 * Ebony Jackson (talk) 20:23, 13 November 2020 (UTC)
 * Personally, I pronounce "naught" and "nought" the same way, though I understand that this may differ from person to person. Anyway, my question is about the spelling, not the pronunciation. Ebony Jackson (talk) 20:32, 13 November 2020 (UTC)

Here are some further references suggesting that when a subscript 0 is spelled out, it is preferable to write "nought" instead of "naught" (and this is not only a matter of British English vs. American English): Ebony Jackson (talk) 07:06, 21 November 2020 (UTC)
 * As a rule of thumb, "nought" (or "aught") is preferred when dealing with numbers, while "naught" is preferred outside of math. -- Prof. Merrill Perlman, Columbia Graduate School of Journalism
 * "nought" is also synonym with "zero" in British English -- Marius Alza
 * Nought is conventionally used in British English for the number zero...In both British English and American English, naught is used in nonmathematical contexts to mean nothing. -- grammarist.com
 * It rubs me badly the wrong way for some reason, but I have to admit you have your cites in order. --Trovatore (talk) 00:05, 22 November 2020 (UTC)

Polish space cardinality
Prior to July 16th, this page stated that "Every uncountable coanalytic subset of a Polish space $$\,X\,$$ has cardinality $$\,\aleph_1\,$$ or $$\,2^{\aleph_0}\,$$". The natural reading of "or" in this context is as "that is to say", which assumes the continuum hypothesis. I noted this and deleted the sentence.

On July 17th, User:JRSpriggs, reverted my edit, seemingly taking "or" in its strict mathematical sense. That is to say "every such subset has cardinality $$\,\aleph_1\,$$ or it has cardinality $$\,2^{\aleph_0}\,$$". That is technically correct, but only because every such set has cardinality $$\,2^{\aleph_0}\,$$" (which is stated explicitly in the referenced article, which uses c for $$\,2^{\aleph_0}\,$$). As such (even rephrased to remove the English ambiguity) this statement is misleading and belongs in the article about the continuum, aka $$\,\beth_1\,$$. (And certainly not under $$\,\aleph_1\,$$).

On August 6th, an anonymous user changed the sentence to "Every uncountable coanalytic subset of a Polish space $$\,X\,$$ has cardinality $$\, \aleph_1\, = \, 2^{\aleph_0}$$" presumably because they read it the same way I did. It is now explicitly assuming the continuum hypothesis.

I'm deleting the sentence. User:JRSpriggs is welcome to revise the sentence and move it to somewhere appropriate. Please do not revert. --Rxtreme (talk) 18:30, 7 August 2021 (UTC)
 * Good removal. Not clear why this point belongs in the article about aleph numbers. --Trovatore (talk) 19:15, 7 August 2021 (UTC)

 * It may be appropriate to remove it on the ground of irrelevance to the article, but I did not like removing a sentence backed by a citation on grounds of nothing more than a misreading of what it said.
 * For example, if the continuum is $$\,\aleph_3\,$$, then the cited article could mean that no subset of the continuum of cardinality $$\,\aleph_2\,$$ can be Polish. This would be a non-trival use of the property of being of cardinality $$\,\aleph_1\,$$ and mentioning it here (which is our only article on $$\,\aleph_1\,$$) would be significant. JRSpriggs (talk) 02:13, 9 August 2021 (UTC)
 * Well, if the continuum is $$\aleph_3$$, then no coanalytic subset of R can have cardinality $$\aleph_1$$, either. The theorem is that any coanalytic subset of a Polish space is either countable or has full cardinality (that is, $$2^{\aleph_0}$$, the same as the original space).  (By the way, this goes way beyond coanalytic, except to prove it for more and more complicated pointclasses, you need more and more large cardinals.) --Trovatore (talk) 05:04, 9 August 2021 (UTC)
 * Strictly speaking this is off-topic, but just in case you're interested, this fact goes through the fact that coanalytic sets have the perfect set property, for which I think Borel determinacy is enough . Search for "unfolded perfect set game".  More large cardinals give you more determinacy, and therefore the perfect set property for more and more complicated pointclasses. --Trovatore (talk) 01:10, 11 August 2021 (UTC)
 * Hmm, actually now I'm not sure. It looks like Borel determinacy gets you the perfect set property for analytic, not for coanalytic.  It could be JR's interpretation was correct after all. --Trovatore (talk) 01:16, 11 August 2021 (UTC)
 * Your interpretation of the sentence was contorted. I think you're the only person to read it that way (I didn't make the following edit, which was clearly for clarification) and it didn't reflect the actual source (so clearly the author of that sentence didn't intend your interpretation). (Reading an article that proves P and writing "P or Q or R" in the Wikipedia article is misleading and absurd.)
 * It's moot at this point, but I wish you considered which reading is more natural, and what the reference says, before reverting my edit.--Rxtreme (talk) 05:30, 10 August 2021 (UTC)
 * Actually JR might be right; this might be the most you can prove about $$\Pi^1_1$$ in ZFC alone. I wouldn't be shocked to find, say, that the reals of L were $$\Pi^1_1$$ in some forcing extension of L that falsifies CH but doesn't collapse $$\aleph_1$$. --Trovatore (talk) 01:53, 11 August 2021 (UTC)
 * Are you claiming that this is in the cited article? I don't see it.--Rxtreme (talk) 05:20, 11 August 2021 (UTC)
 * I'm not sure what you mean by "the cited article". The ref is to a book, one I don't have. --Trovatore (talk) 05:30, 11 August 2021 (UTC)
 * The preview available online does include, on page 20, the statement Proposition 1.4.17 can be generalized to show that each uncountable analytic subset of a Polish space has cardinality $$\mathfrak{c}\ldots$$. But note that it says analytic, not coanalytic.  Because every coanalytic set is the union of $$\aleph_1$$ Borel sets, it would follow that the only possible cardinalities for a coanalytic set are countable, $$\aleph_1$$, or $$2^{\aleph_0}$$.  I don't know whether that's all you can prove in ZFC or not.  Just a little large-cardinal strength would eliminate the $$\aleph_1$$ possibility, unless of course CH holds. --Trovatore (talk) 05:51, 11 August 2021 (UTC)
 * The only mention of coanalytic set in the source is "In fact, Proposition 1.4.13 is a particular case of a deeper theorem in which the same conclusion holds under the more general assumption that the equivalence relation E is a Borel set (or even a coanalytic set) in X × X. A proof of this more general result is given in [145, Theorem 32.1]." There's nothing about aleph-null.--Rxtreme (talk) 21:37, 11 August 2021 (UTC)
 * The source is a book, so it's kind of hard to know what's not in it, but anyway I agree that the snippets we've discussed don't justify the claim. Even if it did, it wouldn't really be a fact about $$\aleph_1$$ as much as a fact about coanalytic sets.  I don't support bringing the text back.
 * Still, I thought it was worth bending the talk-page rules to point out that some of the stuff said in this discussion wasn't really true, so that readers don't internalize misinformation, and to give JR credit where due. --Trovatore (talk) 21:50, 11 August 2021 (UTC)
 * Update: There's a relevant thread at WP:RD/Math, eventually to be archived here.  I haven't checked all the details but, yes, it looks like all you can prove in ZFC alone for coanalytic is indeed "$$\aleph_1$$ or $$2^{\aleph_0}$$", and  seems to have read it correctly after all.  I do not see any indication that this was justified by the claimed source (though again, I don't have the book; I was just going off the online preview), and even if it were, I don't think it would make sense in this article.  I am almost tempted to go through the history and see when it was added and by whom; could have been someone who just plopped it in out of personal knowledge without updating the source. --Trovatore (talk) 16:31, 13 August 2021 (UTC)

definitions don't make sense without more context
If you didn't already know, how would you understand $$\aleph_\omega = \sup\{\aleph_n | n\in \{ 0, 1, 2, ... \}\}$$? or $$\aleph_\lambda = \bigcup_{\beta < \lambda} \aleph_\beta$$? What is a supremum or union of cardinal numbers?

Yes, maybe you happen to know that we often identify a cardinal with its initial ordinal, and that we often identify an ordinal with the set of all smaller ordinals. But maybe it would be better to explain without this convention. And anyway, I don't see this convention explained at all in the article at the moment.

Yes, there is a link to least upper bound, but maybe it would be better to have at least a short explanation in place. And anyway, I don't see anything in the other article about cardinal suprema at all!

For that matter, what is $$\aleph_n$$ for finite n? Yes, $$\aleph_1$$ was explained as the cardinality of the set of all countable ordinals, so maybe a reader would guess what $$\aleph_2$$ means. But still, I think it wouldn't hurt to be clearer. --2607:FEA8:86DC:B0C0:B9B4:5D25:5007:EEE7 (talk) 23:31, 19 January 2022 (UTC)

 * Ah, for $$\aleph_2$$, I missed that there is also a link to successor cardinal. But again, maybe it would be better to have at least a short explanation in-place. And also, the link occurs in a later section of the article than the reference to $$\aleph_2$$. --2607:FEA8:86DC:B0C0:B9B4:5D25:5007:EEE7 (talk) 23:49, 19 January 2022 (UTC)

 * You may take this "The α-th infinite initial ordinal is written $$\omega_\alpha$$. Its cardinality is written $$\,\aleph_\alpha~.$$" as the definition, if that helps. JRSpriggs (talk) 20:02, 20 January 2022 (UTC)

 * It wouldn't hurt to articulate $$\aleph_2$$ as much as possible. This would give the reader a stronger sense of grounding. The current sequence of "0,1, trivial successor function that is obvious to all, omega" is not very intellectually satisfying. It feels like a cop-out or hand-wavey or I've-been-gypped-give-me-my-money-back feel to it. 67.198.37.16 (talk) 03:31, 25 March 2024 (UTC)

In popular culture
In Futurama s.2 ep.12 “Raging Bender” at 01:30 a movie theater is named Loew’s ℵ0-plex — a pun on Multiplex (movie theater). Tuvalkin (talk) 08:13, 22 November 2022 (UTC)

title: Talk:Aleph number


text:
Compiègne

Compiègne (Compiène) is a commune in the Oise department in northern France. It is located on the river Oise. Its inhabitants are called Compiégnois.

Administration
Compiègne is the seat of two cantons:
 * Compiègne-1 (with 19 communes and part of Compiègne)
 * Compiègne-2 (with 16 communes and part of Compiègne)

History by year

 * 665 - Saint Wilfrid was consecrated Bishop of York. Wilfrid refused to be consecrated in Northumbria at the hands of Anglo-Saxon bishops. Deusdedit, Archbishop of Canterbury, had died, and as there were no other bishops in Britain whom Wilfrid considered to have been validly consecrated, he travelled to Compiègne, to be consecrated by Agilbert, the Bishop of Paris.
 * 757 - Byzantine emperor Constantine V sent to Compiègne a gift for Pepin the Short : France's first organ.
 * 833 - Louis the Pious (also known as King Louis I, the Debonair) was deposed in Compiègne.
 * February 888 - Odo, Count of Paris and king of the Franks was crowned in Compiègne.
 * 23 May 1430 - During the Hundred Years' War, Joan of Arc was captured by the Burgundians while attempting to free Compiègne. They then sold her to the English.
 * 1557 - Battle of St. Quentin, where Habsburg Spanish forces defeat a French army in the Italian War of 1551–1559
 * 1558 - The English occupy Compiègne
 * 1624 - Compiègne gave its name to the Treaty of Compiègne, a treaty of alliance concluded by Cardinal Richelieu with the Dutch.
 * 1630 - Marie de' Medici's attempts to displace Richelieu ultimately led to her exile to Compiègne, from where she escaped to Brussels in 1631.
 * 17 July 1794 - The Martyrs of Compiègne are executed in Paris during the Reign of Terror.
 * 1900 - The golf events for the 1900 Summer Olympics took place.
 * 11 November 1918 - The Armistice with Germany (Compiègne), agreed at Le Francport near Compiègne, ends fighting of World War I
 * 22 June 1940 - Another Armistice with France (Second Compiègne) was signed between Nazi Germany and the defeated France in Le Francport, near Compiègne, in the same place as in 1918, in the same railroad carriage, but with the seats swapped.
 * 1941 - During the German occupation of France, the Compiègne internment camp was established in Compiègne. A memorial of the camp, and another along the railway tracks, commemorate the tragedy.
 * 1972 - Creation of the University of Technology of Compiègne
 * 1977 - The starting location of the Paris–Roubaix bicycle race was changed from Paris to Compiègne.

Population
Compiègne is the central commune of an urban unit with 70,699 inhabitants, and a larger commuter zone with 141,504 inhabitants as of 2017. The population data in the table and graph below refer to the commune of Compiègne proper.

Museums

 * Château de Compiègne - the castle itself, and museums of the Second French Empire and of motoring and tourism within its walls
 * Musée Antoine Vivenel
 * Museum of historic figurines
 * Memorial of internment and deportation

Compiègne Forest
The Glade of the Armistice in the Compiègne Forest was the site of the signing of two armistices; those of 11 November 1918 and 22 June 1940. Hitler specifically chose the location of the second, and had the original signing carriage moved from Paris to Compiègne, as an irony for the defeated French.

The site still houses several memorials to the 1918 armistice, including a copy of the original railway carriage. The original, Marshal Foch's Carriage was taken to Germany as a trophy of victory following the second armistice. Various rumors about what happened to this railway-carriage thereafter, have flourished ever since. Some believe it was destroyed by the SS in Thuringia in April 1945; others say this happened in Berlin, but most likely was it destroyed during an allied air-raid on Berlin. The latter version seems most plausible, since Ferdinand Foch's carriage actually was displayed at a Berlin museum.

The University of Technology of Compiègne
Compiègne is home to the University of Technology of Compiègne (UTC), one of the top ranking engineering school in France, founded as a Technology University in 1972 to provide an alternative to the traditional "grandes écoles" for students interested in technologies and applied science.

Transport
Compiègne station offers connections with Paris, Amiens, Cambrai and several regional destinations. The nearest motorway is the A1 Paris-Lille.

Cycling
Since 1977, Compiègne is the traditional start city of the famous Paris–Roubaix bicycle race. It was also the finish city of 3rd stage in the 2007 Tour de France.

Notable people
Compiègne has been home to:
 * Roscellinus (~1050 - ~1122), philosopher and theologian, often regarded as the founder of Nominalism
 * Pierre d'Ailly (1350–1420), theologian and cardinal of the Roman Catholic Church
 * Albert Robida (1848–1926), illustrator, etcher, lithographer, caricaturist, and novelist
 * Eugène Albertini (1880–1941), teacher in Latin literature, historian of ancient Rome, and epigrapher of Latin texts
 * Marcel Tabuteau (1887–1966), Oboist, regarded as the founder of American oboe playing.
 * Suzanne Lenglen (1899–1938), tennis player, international female sport star
 * Lucas Debargue (1990–), pianist and composer who works in both the classical and jazz fields.
 * The Martyrs of Compiègne

International relations
Compiègne is twinned with:

Compiègne is also partnered with:

title: Compiègne