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Lupe Vélez

María Guadalupe Villalobos "Lupe" Vélez (July 18, 1908 – December 14, 1944) was a Mexican actress, singer, and dancer during the Golden Age of Hollywood cinema.

Vélez began her career as a performer in Mexican vaudeville in the early 1920s. After moving to the United States, she made her first film appearance in a short in 1927. By the end of the decade, she was acting in full-length silent films and had progressed to leading roles in The Gaucho (1927), Lady of the Pavements (1928) and Wolf Song (1929), among others. Vélez made the transition to sound films without difficulty. She was one of the first successful Mexican actresses in Hollywood. During the 1930s, her explosive screen persona was exploited in successful comedic films like Hot Pepper (1933), Strictly Dynamite (1934) and Hollywood Party (1934). In the 1940s, Vélez's popularity peaked while appearing as Carmelita Fuentes in eight Mexican Spitfire films, a series created to capitalize on her fiery personality.

Nicknamed The Mexican Spitfire by the media, Vélez's personal life was as colorful as her screen persona. She had several highly publicized romances with Hollywood actors and a stormy marriage with Johnny Weissmuller. Vélez died at age 36 in December 1944 of an intentional overdose of the barbiturate drug Seconal. Her death and the circumstances surrounding it were the subject of speculation and controversy.

Childhood and education
Vélez was born in the city of San Luis Potosí in Mexico, the daughter of Jacobo Villalobos Reyes, a colonel in the armed forces of the dictator Porfirio Diaz, and his wife Josefina Vélez, an opera singer according to some sources, or vaudeville singer according to others. She was one of five children; she had three sisters: Mercedes, Reina and Josefina and a brother, Emigdio. The Villalobos were considered a prominent, financially comfortable family in San Luis Potosí. According to Vélez's second cousin, they lived in a large home, and most of the male members received a college education.

At the age of 13, her parents sent her to study at Our Lady of the Lake (now Our Lady of the Lake University) in San Antonio, Texas, where Vélez learned to speak English and dance. She later admitted that she liked dance class, but was otherwise a poor student.

Beginnings in Mexico and arrival in the United States (1924–26)
Vélez began her career in Mexican revues in the early 1920s. She initially performed under her paternal surname (see Hispanic American naming customs) of Villalobos, but after her father returned home from the war (he did not die in combat as some sources state), he was outraged that his daughter had decided to become a stage performer. She chose her maternal surname Vélez as her stage name. Their mother introduced Vélez and her sister Josefina to the popular Spanish Mexican vedette María Conesa, "La Gatita Blanca". Vélez debuted in a show led by Conesa, where she sang "Oh Charley, My Boy" and danced the shimmy. In 1924, Aurelio Campos, a young pianist and friend of the Vélez sisters, recommended Vélez to stage producers Carlos Ortega and Manuel Castro. Ortega and Castro were preparing a season revue at the Regis Theatre, and hired Vélez to join the company in March 1925. Later that year, Vélez starred in the revues Mexican Rataplan and ¡No lo tapes! (both parodies of the Bataclan's shows in Paris). Her suggestive singing and provocative dancing was a hit with audiences, and she soon established herself as one of the main stars of vaudeville in Mexico. After a year and a half, Vélez left the revue after the manager refused to give her a raise. She then joined the Teatro Principal, but was fired after three months due to her "feisty attitude". Vélez was quickly hired by the Teatro Lirico, where her salary rose to 100 pesos a day.

Vélez, whose volatile and spirited personality and feuds with other performers were often covered by the Mexican press, also honed her ability for garnering publicity. Her most bitter rivals included the Mexican vedettes Celia Padilla, Celia Montalván, and Delia Magaña. Called La Niña Lupe because of her youth, Vélez soon established herself as one of the main stars of vaudeville in Mexico. Among her admirers were notable Mexican poets and writers like José Gorostiza and Renato Leduc.

In 1926, Frank A. Woodyard, an American who had seen Vélez perform, recommended her to stage director Richard Bennett (the father of actresses Joan and Constance Bennett). Bennett was looking for an actress to portray a Mexican cantina singer in his upcoming play The Dove. He sent Vélez a telegram inviting her to Los Angeles to appear in the play. Vélez had been planning to go to Cuba to perform, but quickly changed her plans and traveled to Los Angeles. However, upon arrival, she discovered that she had been replaced by another actress.

While in Los Angeles, she met the comedian Fanny Brice. Brice was taken with Vélez and later said she had never met a more fascinating personality. She promoted Vélez's career as a dancer and recommended her to Flo Ziegfeld, who hired her to perform in New York City. While Vélez was preparing to leave Los Angeles, she received a call from Metro-Goldwyn-Mayer producer Harry Rapf, who offered her a screen test. Producer and director Hal Roach saw Vélez's screen test and hired her for a small role in the comic Laurel and Hardy short Sailors, Beware!

Breakthrough and success (1927–38)

After her debut in the short film Sailors, Beware!, Vélez appeared in the Hal Roach short What Women Did for Me, with Charley Chase. Later that year, she did a screen test for the upcoming Douglas Fairbanks full-length film The Gaucho. Fairbanks was impressed by Vélez and he quickly signed her to a contract. Upon its release in 1927, The Gaucho was a hit and critics were duly impressed with Vélez's ability to hold her own with Fairbanks, who was well known for his spirited acting and impressive stunts.

Vélez made her second major film, Stand and Deliver (1928), directed by Cecil B. DeMille. That same year, she was named one of the WAMPAS Baby Stars. In 1929, Vélez appeared in Lady of the Pavements, directed by D. W. Griffith and Where East Is East, playing a young Chinese woman. In the Western film Wolf Song, directed by Victor Fleming, she appears with Gary Cooper. As she was regularly cast as "exotic" or "ethnic" women that were volatile and hot tempered, gossip columnists took to referring to Vélez as "Mexican Hurricane", "The Mexican Wildcat", "The Mexican Madcap", "Whoopee Lupe" and "The Hot Tamale".

By 1929, the film industry was transitioning from silents to sound films. Several stars of the era saw their careers abruptly end due to heavy accents or voices that recorded poorly. Studio executives predicted that Vélez's accent would probably hamper her ability to make the transition. That idea was dispelled after she appeared in her first all-talking picture in 1929, the Rin Tin Tin vehicle Tiger Rose. The film was a hit and Vélez's sound career was established.

With the arrival of talkies, Vélez appeared in a series of Pre-Code films like Hell Harbor (directed by Henry King), The Storm (1930, directed by William Wyler), and the crime drama East Is West, with Edward G. Robinson (1930). In 1931, she appeared in her second film for Cecil B. DeMille, Squaw Man, with Warner Baxter, and in Resurrection, directed by Edwin Carewe. In 1932, Vélez filmed The Cuban Love Song (1931), with the popular singer Lawrence Tibbett. That same year, she had a supporting role in Kongo (a sound remake of West of Zanzibar), with Walter Huston. She also starred in Spanish-language versions of some of her movies produced by Universal Studios like Resurrección (1931, the Spanish version of Resurrection), and Hombres en mi vida (1932, the Spanish version of Men in Her Life). Vélez found her niche in comedy, playing beautiful, volatile characters.

In February 1932, Vélez took a break from her film career and traveled to New York City where she was signed by Broadway impresario Florenz Ziegfeld, Jr. to take over the role of "Conchita" in the musical revue Hot-Cha!. The show also starred Bert Lahr, Eleanor Powell and Buddy Rogers.

In 1933, Vélez appeared in the films The Half-Naked Truth with Lee Tracy and Hot Pepper, with Victor McLaglen and Edmund Lowe. Later that year, she returned to Broadway where she starred with Jimmy Durante in the musical revue Strike Me Pink. In 1934, she filmed Palooka and Strictly Dynamite (both also with Durante). That same year, Vélez was cast as "Slim Girl" in Laughing Boy with Ramón Novarro. The film was quietly released and largely ignored. The few reviews it received panned the film, but praised Vélez's performance. She had more success with her brief appearance in the all-comedy-cast film Hollywood Party, where she has an egg-breaking routine with Laurel and Hardy. Although Vélez was a popular actress, MGM Studios did not renew her contract in 1934. Over the next few years, Vélez worked for various studios as a freelance actress; she also spent two years in England where she filmed The Morals of Marcus and Gypsy Melody (both 1936). She returned to Los Angeles the following year where she appeared in the final part of the Wheeler & Woolsey comedy High Flyers (1937).

Vélez last Broadway performance was in the 1938 musical You Never Know, by Cole Porter. The show received poor reviews from critics, but received a large amount of publicity due to the feud between Vélez and fellow cast member Libby Holman. Holman also was irritated by the attention Vélez garnered from the show with her impressions of several actresses, including Gloria Swanson, Katharine Hepburn and Shirley Temple. The feud came to a head during a performance in New Haven, Connecticut after Vélez punched Holman between curtain calls and gave her a black eye. The feud effectively ended the show.

Upon her return to Mexico City in 1938 to star in her first Mexican film, Vélez was greeted by ten thousand fans. The film La Zandunga directed by Fernando de Fuentes, co-starring Mexican actor Arturo de Córdova, was a critical and financial success and Vélez was slated to appear in four more Mexican films. She instead returned to Los Angeles and went back to work for RKO.

Continued success and final roles (1939–44)

In 1939, Vélez was cast with Leon Errol and Donald Woods in The Girl from Mexico. Despite being a B film, it was a hit with audiences and RKO re-teamed her with Errol and Woods for the sequel Mexican Spitfire. That film was also a success and led to a series of Spitfire films (eight in all). In the series, Vélez portrays Carmelita Lindsay, a temperamental yet friendly Mexican singer married to Dennis "Denny" Lindsay (Woods), an elegant American gentleman. The Spitfire films rejuvenated Vélez's career. Moreover, they were films in which a Latina headlined for eight movies straight –a true rarity.

In addition to the Spitfire series, she was cast in other musical and comedy features for RKO, Universal Pictures, and Columbia Pictures. Some of these films were Six Lessons from Madame La Zonga (with Leon Errol, 1941), Playmates (with John Barrymore, 1941), and Redhead from Manhattan (1943). The last film has Vélez playing the dual role of a Broadway star and her visiting twin sister, and offers the viewer a rare chance to hear Lupe Vélez's actual speaking voice. She plays the visiting sister with her customary, exaggerated Mexican accent, but plays the actress with her own, gently inflected voice, betraying only a trace of an accent.

In 1943, the final film in the Spitfire series, Mexican Spitfire's Blessed Event, was released. By that time, the mistaken-identity scripts and situations had been repeating themselves, and the novelty of the series had begun to wane, but Vélez's energy and Errol's clowning never flagged.

Vélez co-starred with Eddie Albert in a 1943 romantic comedy, Ladies' Day, about an actress and a baseball player. In 1944, Vélez returned to Mexico to star in an adaptation of Émile Zola's novel Nana, which was well received. It would be her final film. After filming wrapped, Vélez returned to Los Angeles and began preparing for another stage role in New York.

Public image and personality
Throughout her career, Vélez's onscreen persona of a hot-tempered, lusty "wild" woman was closely tied to her off-screen personality. The press often referred to her by such names as "The Mexican Spitfire", "The Mexican It girl" and "The Mexican Kitten". Publicly promoted with the "Whoopee Lupe" persona that tried to define her, she dismissed the idea that she was uncontrollably wild. In an interview, she said:

"What I attribute my success?, I think, simply, because I'm different. I'm not beautiful, but I have beautiful eyes and know exactly what to do with them. Although the public thinks that I'm a very wild girl. Actually I'm not. I'm just me, Lupe Vélez, simple and natural Lupe. If I'm happy, I dance and sing and acted like a child. And if something irritates me, I cry and sob. Someone called that 'personality'. The Personality is nothing more than behave with others as you really are. If I tried to look and act like Norma Talmadge, the great dramatic actress, or like Corinne Griffith, the aristocrat of the movies, or like Mary Pickford, the sweet and gentle Mary, I would be nothing more than an imitation. I just want to be myself: Lupe Vélez."

Vélez's off-screen behavior blurred the line between her onscreen persona and her real personality. After her death, journalist Bob Thomas recalled that Vélez was a "lively part of the Hollywood scene" who wore loud clothing and made as much noise as possible. She attended boxing matches every Friday night at the Hollywood Legion Stadium and would stand on her ringside seat and scream at the fighters.

Vélez's temper and jealousy in her often tempestuous romantic relationships were well documented and became tabloid fodder, often overshadowing her career. Vélez was straightforward with the press and was regularly contacted by gossip columnists for stories about her romantic exploits. One such incident included Vélez chasing her lover Gary Cooper around with a knife during an argument and cutting him severely enough to require stitches. After their breakup, Vélez attempted to shoot Cooper while he boarded a train. During her marriage to actor Johnny Weissmuller, stories of their frequent physical fights were regularly reported in the press. Vélez reportedly inflicted scratches, bruises, and love-bites on Weissmuller during their fights and "passionate love-making".

Vélez often targeted fellow actresses whom she deemed rivals, professionally or otherwise, a habit which began back in her vaudeville days and continued in films. Vélez's image was that of a wild, highly sexualized woman who spoke her mind and was not considered a "lady", while fellow Mexican actress Dolores del Río projected herself as sensual, but elegant and restrained, often hailing from aristocratic roots. Vélez hated del Río, and called her a "bird of bad omen". Del Río was terrified to meet her in public places. When this happened, Vélez was scathing and aggressive. Vélez openly mimicked del Río, ironically making fun of her elegance. Vélez also disliked Marlene Dietrich whom she suspected of having an affair with Gary Cooper while filming Morocco in 1930. Her rivalries with Jetta Goudal, Lilyan Tashman and Libby Holman were well documented. In retaliation, Vélez would perform scathing impersonations of the women she disliked at Hollywood parties. Also notable are her imitations of figures such as Greta Garbo, Marlene Dietrich, Fanny Brice, Gloria Swanson, Katharine Hepburn, Simone Simon, and Shirley Temple.

Relationships and marriage
Vélez was involved in several highly publicized and often stormy relationships. Upon arriving in Los Angeles, she was linked to actors Tom Mix, Charlie Chaplin and Clark Gable. Her first long-term, high-profile relationship was with Gary Cooper. Vélez and Cooper met while filming 1929s Wolf Song and began a two-year relationship that was passionate and often stormy. When angered, Vélez was reported to have physically assaulted Cooper. Cooper eventually ended the relationship in mid-1931, at the behest of his mother Alice who after meeting her, strongly disapproved of Vélez. With plans to marry him gone, she spoke to the press in 1931: "I turned Cooper down because his parents didn't want me to marry him and because the studio thought it would injure his career. Now its over, I'm glad I feel so free ... I must be free. I know men too well they are all the same, no? If you love them they want to be boss. I will never have a boss." The rocky relationship had taken its toll on Cooper, who had lost 45 pounds and was suffering from nervous exhaustion. Paramount Pictures ordered him to take a vacation to recuperate and while he was boarding the train, Vélez showed up at the station and fired a pistol at him.

After her breakup with Cooper, Vélez began a short-lived relationship with actor John Gilbert. They began dating in late 1931 while Gilbert was separated from his third wife Ina Claire. Rumors of an engagement were fueled by the couple, but Gilbert ended the relationship in early 1932, and attempted to reconcile with Claire.

Shortly thereafter, Vélez met Tarzan actor Johnny Weissmuller while the two were in New York. They dated off and on when they returned to Los Angeles, while Vélez also dated actor Errol Flynn. On October 8, 1933, Vélez and Weissmuller were married in Las Vegas. There were reports of domestic violence and public fights. In July 1934, after ten months of marriage, Vélez filed for divorce citing "cruelty". She withdrew the petition a week later after reconciling with Weissmuller. On January 3, 1935, she filed for divorce a second time and was granted an interlocutory decree. That decree was dismissed when the couple reconciled a month later. In August 1938, Vélez filed for divorce for a third time, again charging Weissmuller with cruelty. Their divorce was finalized in August 1939.

After the divorce became final, Vélez began dating actor/polo player Guinn "Big Boy" Williams in late 1940. The couple were engaged, but never married. In late 1941, she became involved with author Erich Maria Remarque. Actress Luise Rainer recalled that Remarque told her "with the greatest of glee" that he found Vélez's volatility wonderful when he recounted to her an occasion where Vélez became so angry with him that she took her shoe off and hit him with it. After dating Remarque, Vélez was linked to boxers Jack Johnson and Jack Dempsey.

In 1943, Vélez began an affair with her La Zandunga co-star Arturo de Córdova. De Córdova had recently moved to Los Angeles after signing with Paramount. Despite the fact that De Córdova was married to Mexican actress Enna Arana with whom he had four children, Vélez granted an interview to gossip columnist Louella Parsons in September 1943 and announced that the two were engaged. She told Parsons that she planned to retire after marrying De Córdova to "cook ... and keep house". Vélez ended the engagement in early 1944, after de Córdova's wife refused to give him a divorce.

Vélez then met and began dating a struggling young Austrian actor named Harald Maresch, whose stage name was Harald Ramond. In September 1944, she discovered she was pregnant with Ramond's child. She announced their engagement in late November 1944. On December 10, four days before her death, Vélez announced she had ended the engagement and kicked Ramond out of her home.

Death
On the evening of December 13, 1944, Vélez dined with two friends, the silent-film star Estelle Taylor and Venita Oakie. In the early morning hours of December 14, Vélez retired to her bedroom, where she consumed 75 Seconal pills and a glass of brandy. Her secretary, Beulah Kinder, said that she found the actress's body on her bed later that morning. A suicide note addressed to Harald Ramond was found nearby. It read:

"To Harald, May God forgive you and forgive me too, but I prefer to take my life away and our baby's before I bring him with shame or killing him. – Lupe."

On the back of the note, Vélez wrote:

"How could you, Harald, fake such a great love for me and our baby when all the time, you didn't want us? I see no other way out for me, so goodbye, and good luck to you, Love Lupe."

The day after Vélez's death, Harald Ramond told the press that he was "so confused" by Vélez's suicide, and claimed that even though the two had broken up, he had agreed to marry Vélez. He admitted that he once asked Vélez to sign an agreement stating that he was only marrying her to "give the baby a name", but claimed he only did so because he and Vélez had had a fight, and he was in a "terrible temper". Actress Estelle Taylor, who was with Vélez from 9:00 the previous night until 3:30 the morning Vélez died, told the press that Vélez had told her of her pregnancy, but said she would rather kill herself than have an abortion. Beulah Kinder, Vélez's secretary, later told investigators that after Vélez broke off the relationship with Ramond, she planned to go to Mexico to have her baby. Kinder said Vélez soon changed her mind after concluding that Ramond "faked" the relationship and considered having an abortion.

The day after Vélez's death, the Los Angeles County coroner requested that an inquest be opened to investigate the circumstances surrounding her death. On December 16, the coroner dropped the request, after determining that Vélez had written the notes, and that she had intended to kill herself. On December 22, a funeral for Vélez was held at the mortuary at Forest Lawn Memorial Park in Los Angeles. Among the pallbearers were Vélez's ex-husband, Johnny Weissmuller, and actor Gilbert Roland. After the service, Vélez's body was sent by train to Mexico City, where a second service was held on December 27. Her body was then interred at Panteón Civil de Dolores Cemetery.

Alternative theories and urban legend
Despite the coroner's ruling that Vélez committed suicide to avoid the shame of bearing an illegitimate child, some authors have speculated that this was not entirely true.

In the book From Bananas to Buttocks: The Latina Body in Popular Film and Culture, Rosa-Linda Fregoso wrote that Vélez was known for her defiance of contemporary moral convention, and that it seems unlikely that she could not have reconciled having a child out of wedlock. Fregoso believes that in the final year of her life, Vélez exhibited signs of extreme mania and depression. Fregoso goes on to speculate that Vélez's death may have been the result of an untreated mental illness such as bipolar disorder.

Robert Slatzer (who later claimed to have been secretly married to Marilyn Monroe) claimed that a few weeks before Vélez's death, he interviewed her at her home, and she confided in him that she was pregnant with Gary Cooper's child (by that time, Cooper was married to socialite Veronica "Rocky" Balfe). According to Slatzer, Vélez said that Cooper refused to acknowledge the child, believing that Harald Ramond was the father. After Vélez died, Slatzer said he asked Cooper about the situation, and Cooper confirmed that it was possible he might have been the father. Slatzer further claimed that he also interviewed Clara Bow (who had also dated Cooper in the 1920s), who revealed that shortly before Vélez's death, Cooper called her and screamed that he was going to kill Harald Ramond for impregnating Vélez. Slatzer claimed that Bow told him that she never believed Vélez's baby was fathered by Ramond, and that she was convinced that Vélez had attempted to get Ramond to marry her to protect Cooper's reputation. Biographer Michelle Vogel speculated that if Cooper was the father, his rejection of Vélez and their child, coupled with the idea of having to raise a child alone, may have sent Vélez "over the edge".

In the 2002 book Tarzan, My Father, Johnny Weissmuller, Jr., recounted the events surrounding Vélez's death as a mystery caused by an attempt to "put a lid" on what happened. It states her housekeeper discovered her body and called Bo Roos, Vélez's business manager, who called his friend and Beverly Hills Police Chief Anderson to the scene. The book states after Vélez arranged to meet Ramond, decorated her room, and dressed in a negligee, her ingestion of Seconal was either to calm her nerves to meet him or a failed dramatic gesture to scare him. The book also suggested the baby was fathered possibly by Cooper, not Ramond.

Vélez's death was recounted by Kenneth Anger in his book Hollywood Babylon, and has become urban legend. In Anger's telling, Vélez planned to stage an elaborate suicide scene atop her satin bed, but the Seconal did not mix well with the "Mexi-Spice Last Supper" she had eaten earlier that evening. As a result, she became violently ill, stumbled to the bathroom to vomit, slipped on the bathroom floor tile, and fell head first into the toilet, where she subsequently drowned. Anger claimed that Vélez's "chambermaid" Juanita found her the next morning. Despite the fact that his version of events contradicts published reports and the official ruling, his story is often repeated as fact or for comedic effect – it was recounted in the pilot episode of the television comedy series Frasier, "The Good Son"; referred to in an episode of the cartoon The Simpsons; and mentioned in the song "I Wanna O.D." by the Demolition Doll Rods. Vélez's biographer, Michelle Vogel, points out that it would have been "virtually impossible" for Vélez to have "stumbled to the bathroom" or even get off her bed after having consumed such a large amount of Seconal. Seconal, a barbiturate, is noted for being fast acting even in small doses, and Vélez's death was probably instantaneous. Her death certificate lists "Seconal poisoning" due to "ingestion of Seconal" as the cause of death, not drowning. Further, there was also no evidence to suggest Vélez had vomited. In 2013, the first publication of a police photo of Vèlez's body on the floor of her bedroom was published.

Legacy
For her contribution to the motion picture industry, Vélez has a star on the Hollywood Walk of Fame, at 6927 Hollywood Boulevard.

In 2017, artist Emilio Borjas's sculpture in honor of Vélez was placed in the Garden of San Sebastian, the neighborhood where the actress was born in San Luis Potosí, Mexico.

Books

 * In Robert Stone's novel Children of Light, a troubled actress refers to a suicide-by-drowning scene she has just acted out as "Lupe Vélez takes a dunk".
 * Richard Kadrey also mentions Vélez in the Sandman Slim novel Killing Pretty with reference to the diner, Lupe's in Elysian Park, named after her as the place where she had her last dinner.

Music

 * The group Il Palco della Musica on their 2012 album Tantipiccolipezzi has a song "Lupe Vèlez" (M. Tucci / S. Scotto).

Films and television

 * In 1949, the Los Angeles Daily News reported that the Puerto Rican dancer Marquita Rivera was chosen to star in a biographical film based on the life of the actress. However, due to the controversy over Vélez's suicide at that time, the film was never produced. The Cuban rumbera Amalia Aguilar was also in talks to star in a film about Vélez, but later decided not to work in Hollywood, and returned to Mexico. Latina actress Estelita Rodriguez, who had once played Spitfire-like ingenues for Republic Pictures, announced that she would star in a screen biography of Vélez, but the project died with Rodriguez in March 1966.
 * Andy Warhol's underground film, Lupe (1965), starring Edie Sedgwick as Vélez, is loosely based on the night of her suicide. The film features Sedgwick (in her last film role for Warhol) preparing a "beautiful suicide" only to end up drowning in the toilet bowl.
 * James Leer (Tobey Maguire), while listing Hollywood suicides, says that Lupe Vélez died from taking "a lot of pills" in the 2000 film Wonder Boys.
 * In August 2009, the short film Forever Lupe premiered at the Seattle Latino Film Festival. Directed by Martín Caballero, the film is based on the life of Vélez and features Mexican actress Marieli Romo as Vélez.
 * In 2012, it was reported that Mexican director Carlos Carrera was preparing to film the life of Lupe Vélez in a Mexican-American production. Mexican actress Ana de la Reguera was chosen to play Vélez.
 * Vélez is played by the Cuban-Venezuelan actress María Conchita Alonso in the 2014 film Return to Babylon.
 * In the pilot episode of the sitcom Frasier, "The Good Son", Frasier Crane's producer, Roz Doyle, tries to improve Frasier's outlook on his life by telling him the story of Lupe Vélez, who she says "decided to take one final stab at immortality". Roz re-tells the urban legend version of Vélez's suicide, reminding Frasier that "even though things might not happen like we planned, they can work out anyway". When Frasier asks "how it worked out for Lupe", Roz tells him that "all she wanted was to be remembered. Will you ever forget that story?" Later in the episode, Frasier asks a caller to his radio show: "Have you ever heard of the story of Lupe Vélez?"
 * In the 1997 The Simpsons episode "Homer's Phobia", guest star John Waters takes the Simpson family, sans Homer, on a driving tour of Springfield's shopping district. During the tour, he points out a store where he claims Vélez bought the toilet she drowned in.

Other media

 * Lupe Vélez was featured on the initial issue (no. 1) of the Italian puzzle weekly La Settimana Enigmistica, issued on January 23, 1932. This initiated the tradition, still continued presently by the publisher of the magazine, of featuring the picture of a star on the front page puzzle: a woman and a man on alternate weeks.

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

title: Lupe Vélez


text:
Talk:Dedekind cut

Early comments
Hmmm. I don't see this as a stub. There's more to be added certainly, but the basic facts are here. Andrewa 10:30 16 Jul 2003 (UTC)

I've just added some definitions for comparison, arithmetic operations, and evaluations of supremum and infimum. These definitions are not all from the same source (though presumably all "public knowledge", "well-known" or "sufficiently obvious" ); they may not be entirely consistent with each other and/or with the definitions that were stated (and presently remain unaltered) at the beginning of the article. Please comment, and correct as appropriate. (I've put the same request on Peer review.) Best regards, Frank W ~@) R 02:39, 8 May 2004 (UTC)

Suggestion

 * Start off by defining Dedekind cut in the the simplest case: as a dedeckind cut in the rationals.

 * insert a picture

 * Leave generalizations for later in the article. CSTAR 19:55, 9 Jul 2004 (UTC)

 * (William M. Connolley 21:40, 9 Jul 2004 (UTC)) Probably a good idea. Most of us don't knwo what posets are, Charles.
 * Most may not, but they are pretty ordinary objects in math (e.g. they are studied routinely in lower-division discrete math classes), and so cuts in posets aren't extraordinarily "esoteric" (a relative term), so it's hardly out of place to give the full poset presentation here. Revolver 15:37, 15 Dec 2004 (UTC)

Dodgy (would "imprecise" be more NPOV?) language
(William M. Connolley 21:40, 9 Jul 2004 (UTC)) In several places the article seems to take for granted the existence of the reals. For example, it talks about being able "to do arithmetic on cuts, just like you can do on the reals" (I paraphrase). But (as far as I can understand this) that's the wrong way round. The reals *are* dedekind cuts (or some other definition starting from the rationals). Arithmetic on reals is *defined* by the corresponding actions on cuts. Just as arithmetic on rationals is defined by the corresponding operations on sets of pairs.

Not entirely. One can introduce the reals in a number of ways (e.g. Cauchy sequences of rationals). The main result is that a Dedekind cut of the reals must be at some real x: so, (-∞,x) and [x,+∞) or the other one with x in the left interval. So, there is no 'gap' remaining to isolate within the reals.

I'd be quite happy to move all the general order theory down to the end; and get the Dedekind-story at the top.

Charles Matthews 22:12, 9 Jul 2004 (UTC)

 * (William M. Connolley 22:34, 9 Jul 2004 (UTC)) Err, far be it from me to disagree, but I shall. You too are speaking as if the reals have some independent existence. No matter how you introduce the reals (Cauchy or whatever) their arithmetic properties are defined by the properties of the rationals in the definition; which of course end up being the properties you expect or things would have gone badly wrong...

Actually that's not the approach always adopted. One can just write down axioms for the real numbers qua complete field. People hope there is essentially one model; there might be no model (sceptical, finitist approach), or too many models to handle (let in the logicians with the continuum hypothesis etc.). It doesn't matter so much in practice - one just uses the axioms one has responsibly, and it would be 'bad luck' if there were no model at all. This, IIRC, is how Dieudonne's Foundations of Modern Analysis starts out. Charles Matthews 09:38, 10 Jul 2004 (UTC)

 * WMC, all of the ways Charles mentioned above to construct the reals do not assume any properties of existence of the reals. When he says, "the main result is that a Dedekind cut of the reals must be at some real x", he is speaking about Dedekind cuts of reals. Remember, the reals form a poset, too! So, once you have constructed them, it's perfectly consistent to form Dedekind cuts of reals, and say things about them. Revolver 15:37, 15 Dec 2004 (UTC)

Proposal for simplification
(William M. Connolley 20:05, 11 Jul 2004 (UTC)) I'd like to propose switching the definition to that used at http://planetmath.org/encyclopedia/DedekindCuts.html. This is essentailly that instead of calling (A,B) a cut, we call B the cut. This has the virtue of removing a whole pile of stuff relating to "the other half" which is never interesting. It allows ordering by simple subsetting, which seems a huge virtue. And its logically identical (at least in the context of the reals - CM may know otherwise for more esoteric bases). I think we should also point out somewhere that one can *define* the reals as the cuts (whilst also, of course, noting that there are other ways).

 * Arguably the business of Dedekind cuts of the rationals giving one the reals should be at construction of real numbers, which currently has no detail to speak of. Before changing the definition, and remembering that WP is not supposed to innovate on terminology, one should hesitate a moment, anyway. The point of the definition in terms of order theory is that it is a monotone mapping to the two-element ordered set {0,1} with 0 < 1. Well, that is no more and no less complicated than propositions taking values in the Booleans {yes, no}; and one can equally replace those by the set on which the proposition takes value 'no'. In practice one needs to be fluent in saying things either way. So we should have all that on this page, in some form; as I said, I think the axiomatics can be elsewhere (cf. the approach on the real number page, in fact). Charles Matthews 07:53, 12 Jul 2004 (UTC)

OK, I have now moved the detailed definitions for operations on reals to construction of real numbers. I think that will make it easier to shuffle this page into some better logical order. Charles Matthews 14:59, 12 Jul 2004 (UTC)

William M. Connolley: I'd like to propose switching the definition
 * If the phrase "Dedekind cut" applies to distinct notions or versions (e.g. "original", "simplified", and/or "instead of calling (A,B) a cut, we call B the cut", etc. etc.) then I'd suggest that they should all be covered (eventually);
 * perhaps not necessarily on different pages, but at least in distinct sections, for separate reference;
 * and surely not omitting version(s) "due to Dedekind" and/or "particular representative of a Schnitt (German)"
 * Frank W ~@) R 03:47, 14 Jul 2004 (UTC)

Charles Matthews: I have now moved the detailed definitions for operations on reals to construction of real numbers
 * On one hand that's a good move because that's where the detailed definitions for arithmetic, set, and order operations on Dedekind cuts would seem most relevant. (But why did the move mangle the careful formatting? ...)
 * On the other hand however, while (any and all versions of) "Dedekind cut" can be defined without requiring the detailed definitions for operations, it may well be that different such definitions go with distinct versions of "Dedekind cut". (See also my note from 02:39, 8 May 2004 (UTC), above). Referencing between construction of real numbers and Dedekind cut should have to be correspondingly precise, and edits on either page (ideally) synchronized.
 * Frank W ~@) R 03:47, 14 Jul 2004 (UTC)

The formatting - I just made it standard for this site - it was extraordinarily hard to edit as it was. I don't think we need various different versions of 'Dedekind cut', unless and until they fulfil a clear need. Charles Matthews 07:32, 14 Jul 2004 (UTC)

 * Again, WMC, I think you're confusing the order theory def with the particular case of Q. One of the reasons the case with Q is so important is that one can transfer and expand the field operations on Q into Dedekind cuts of rationals. In other words, the "distinct version of Dedekind cut" you're referring to about the rationals is able to happen not just because we have the poset Q, it's the poset and field Q, altogether, the ordered field. That is precisely why Charles thinks the drawn-out details of this particular construction from Q to R belong at a separate page -- because the field operations are so essential, and field operations aren't part of the theory of general Dedekind cuts in posets. Revolver 15:37, 15 Dec 2004 (UTC)

Definition
There is a problem with the definition: "In mathematics, a Dedekind cut, named after Richard Dedekind, is a partition of the rational numbers into two non-empty parts A and B, such that all elements of A are less than all elements of B, and A contains no greatest element. Dedekind cuts are one method of construction of the real numbers."

It states "non-empty 'parts' A and B" and then goes on to talk about these "parts" as if they are sets, which indeed they are. I will correct this so that it states "sets" and not parts, because only sets have elements by definition. 197.79.9.234 (talk) 08:27, 19 May 2015 (UTC)

 * Yes, thanks IP. It was implicit in the wikilinking of "partition", but it's obviously better to use "sets". I've wikilinked "sets". "Subsets" perhaps? c1cada (talk) 12:03, 19 May 2015 (UTC)

 * There are still more errors:

More generally, a Dedekind cut is a partition of a totally ordered set into two non-empty parts A and B, such that A is closed downwards (meaning that for all a in A, x ≤ a implies that x is in A as well) and B is closed upwards, and A contains no greatest element.

If you are responsible for this article, you might want to add that the lower cut is a Cauchy sequence. 197.79.0.247 (talk) 14:31, 21 May 2015 (UTC)

 * Not responsible for the article . My only contribution here was the last paragraph of the lede. I'm not aware that the definition is seriously in error. I was brought up on Whittaker and Watson myself and the treatment here looks much the same as theirs. I encourage you simply to correct any errors where you see them. As for Cauchy sequence stuff that's probably best dealt with in Construction of the real numbers where there is a section on Dedekind cuts stressing your remarks above. What I would like to see in this article is a discussion of the historical context of Dedekind's construction, including reservations expressed at the time about its perceived geometrical bias, and then perhaps a really careful description of the construction, following perhaps Hardy's celebrated account. Incidentally, as far as getting your concerns looked at by editors here, it would be best to start a new section at the end of the Talk page. Otherwise they are quite likely not to be noticed. c1cada (talk) 15:59, 21 May 2015 (UTC)

 * I am not interested. In my opinion, neither D. Cuts nor Cauchy sequences define the "real" numbers because irrational numbers don't exist. http://www.spacetimeandtheuniverse.com/math/4507-0-999-equal-one-317.html#post21409 197.79.0.85 (talk) 01:45, 23 May 2015 (UTC)
 * Well, yes. You might like to read Hardy's classic account in "A Course of Pure Mathematics" ( first year study in his time at Cambridge around 1910 and still very readable) where he discusses exactly that point of view. c1cada (talk) 03:10, 23 May 2015 (UTC)

 * Have read it and it's nonsense. 197.79.3.185 (talk) 05:26, 14 June 2015 (UTC)

Now I'm getting confused on the definition. Taken at face value, it is true, the standard way of defining addition is not closed. This is a problem. It's okay if we restrict the lower subset not to have an upper bound, e.g. This is how I usually see the definition in analysis books. Now, I looked up in an order theory book, and they disagreed with the definition given here. Basically, their definition was in terms of Galois connections, where the Galois mappings are given (in words) by "take the set of all things bigger than or equal to all things in A" (denote A-up) and similarly for smaller than. This is a Galois connection on the power set, and a Dedekind cut is defined to be a "Galois subset", i.e. a subset so that after you apply each mapping one by the other, you get the same thing. Example: A = (−∞, 0], then A-up = [0, ∞) and (A-up)-down = (−∞, 0] = A. NOTICE: if we change the order of the mappings, we get intervals going to ∞ rather than −∞, but they are still closed. If we think of "identifying" these intervals [a, ∞) with their complements, we get what we think of as open intervals down to −∞, but these complements aren't Galois! (in either order!) I think this is where the confusion about the definition lies. You have to fix the order of the mappings, it seems. By using the "closed-down", "closed-up" mappings, you get both defintions (corr. to different orders of mappings) thrown together instead of just one. If you restrict to just one order of the Galois mappings, then suddenly addition is closed. This leads me to think maybe we have the definition of a Dedekind cut slightly wrong altogether. Revolver 16:28, 15 Dec 2004 (UTC)

 * It's not okay to restrict the lower subset not to have an upper bound, because that is required in the definition. What you see and what you understand from "analysis books" is open to varying interpretations. 197.79.0.247 (talk) 14:33, 21 May 2015 (UTC)

Why is there no mention of the upper set B in the definition? Dutugamunu (talk) 14:24, 14 May 2019 (UTC)

The above suggestion "you might want to add that the lower cut is a Cauchy sequence" is completely wrong. It is most definitely not a Cauchy sequence, especially because it is not a sequence. — Preceding unsigned comment added by 2601:200:C082:2EA0:7D50:88F7:2D96:18EC (talk) 19:33, 20 January 2024 (UTC)

Ordering Dedekind cuts
From the article:
 * "[...] is a Dedekind cut we could call ( −∞, a ); by identifying a with it, the linearly ordered set S is embedded in the set of all Dedekind cuts of S. If the linearly ordered set S does not enjoy the least-upper-bound property, then the set of Dedekind cuts will be strictly bigger than S."

If we do identify (-inf, a) with a, then we are missing all the cuts (-inf, a] -- so the set of Dedekind cuts is bigger than S even if S "enjoys" the l-u-b property. --SirJective 84.151.225.29 15:25, 20 May 2005 (UTC)

 * There is no B for which {(−∞,a],B} is a Dedekind cut - the first set cannot have a greatest element. --Fibonacci 03:06, 10 December 2005 (UTC)

Definition changed
I've changed the definition here and at Construction of real numbers to incorporate the condition that the low/left set contain no greatest element. This is a requirement made by Walter Rudin, MathWorld, PlanetMath, any other source I have ever seen, and most importantly, it is needed to make the construction of R to actually work. Otherwise, as that article pointed out itself, the cuts would not be closed under addition. -- Jao 13:34, Jun 24, 2005 (UTC)

Embedding of a set in the set of its Dedekind cuts

 * "If a is a member of S then the set

 * $$\{ \{ x\in S: x < a \}, \{ x\in S: x \ge a \} \}$$

 * is a Dedekind cut we could call ( −∞, a ); by identifying a with it, the linearly ordered set S is embedded in the set of all Dedekind cuts of S."

Ehm... not quite. What if the left set of that "cut" is empty (so that (A,B) is not a partition of S), or has a greatest element? It wouldn't be a Dedekind cut, would it? The simplest example I can think of is S = {a,b}, with a<b - the set of its Dedekind cuts is empty, so there can be absolutely no embedding of S in it. --Fibonacci 03:13, 10 December 2005 (UTC)

Update: I have deleted the following paragraph from the article, for the reasons stated above:

 * If a is a member of S then the set

 * $$\{ \{ x\in S: x < a \}, \{ x\in S: x \ge a \} \}$$

 * is a Dedekind cut we could call ( −∞, a ); by identifying a with it, the linearly ordered set S is embedded in the set of all Dedekind cuts of S. If the linearly ordered set S does not enjoy the least-upper-bound property, then the set of Dedekind cuts will be strictly bigger than S; conversely, if S has the least-upper-bound property, the set of its Dedekind cuts is order isomorphic to S, by identifying each cut (A,B) with the supremum of A.

axiom of choice
I didn't explain the axiom of choice thing last time, sorry. It was worded poorly. Dedekind was able to prove the completeness of the real numbers without using the axiom of choice by using cuts. This is the significance of Dedekind cuts in Mathematics. The construction using Cauchy sequences requires the proof to use the AoC. Benandorsqueaks 03:33, 17 December 2005 (UTC)

User:Fredrik's Deletion
I am really surprised at User:Fredrik's deletion. Here we had a clear, concise explanation of a Dedekind Cut, verified by a key quotation by Richard Dedekind himself. You couldn't ask for a more apropos account of the meaning of Dedekind's cut. The quotation was the one place in Dedekind's essay that the famous mathematician clearly and distinctly showed, in a few words, how the cut resolved the continuum problem by placing a rational or irrational number at every point on the number line. User:Fredrik doesn't see this. The readers of the article are the losers here. Let the article remain as User:Fredrik wants it, without an enlightening and concise explanation which was accompanied by a very pertinent and important quotation by the renowned mathematician. I reproduce it below for the benefit of curious readers who may judge for themselves the value of the paragraph and quotation:

The Dedekind cut resolves the contradiction between the continuous nature of the number line continuum and the discrete nature of the numbers themselves. Wherever a cut occurs and it is not on a real rational number, an irrational number (which is also a real number) is created by the mathematician. Through the use of this device, there is considered to be a real number, either rational or irrational, at every point on the number line continuum, with no discontinuity. Lestrade 00:38, 27 August 2006 (UTC)Lestrade

 * There is no room in the Dedekind Cut article for Dedekind's quote, which precisely explains the cut. But, on Wikipedia, there is plenty of room, for example, to include such things as a reference in The Simpsons in a Henry Kissinger article. Lestrade 00:04, 1 October 2006 (UTC)Lestrade

 * Personally, I don't see the point of saying that the irrational numbers are "created" by mathematicians. For that matter, all numbers are "created" by mathematicians.  There's nothing more concrete about the number 1 than there is about the number pi.  The whole introduction seems to be rather lacking in understanding of modern math.  Gandalf (talk) 17:43, 24 November 2008 (UTC)

This is a typical lack of understanding of Dedekind's own actual words. What does being "concrete" have to do with positing the existence of a number? Who cares? Ours is an ignorant time and we are proud of our ignorance and our youthful superiority to the old thinkers. Would Gandalf favor us with a clear explanation of his opinion that "The whole introduction seems to be rather lacking in understanding of modern math" or is it so obvious that we should know without being told?Lestrade (talk) 20:26, 24 November 2008 (UTC)Lestrade

 * I beg your pardon. The comment was in haste.  My complaint is mainly against the second paragraph, not the entire introduction.  The sentence "The Dedekind cut resolves the contradiction between the continuous nature of the number line continuum and the discrete nature of the numbers themselves." does not make sense mathematically.  What is the difference between the continuum of numbers and the "numbers themselves"?  The set of numbers being discussed is ambiguous.  If the sentence is referring to the set of rational numbers, it is incorrect because that is not a discrete set as is claimed.  If the set being discussed is the set of integers (which is a discrete set) then the formulation of the reals via Dedekind cuts is incorrect.  I think the paragraph should be deleted due to its lack of mathematical foundation.  Gandalf (talk) 15:26, 26 November 2008 (UTC)

Second paragraph POV (heh)
The second paragraph seems to me to imply the use of Dedkind cuts is the unique way to define (or "create" if you will) the irrational numbers, thus neglecting the method of cauchy sequences. We ought to be clear that to in modern mathematics this is only one possible approach. -- SCZenz 15:16, 8 March 2007 (UTC)

 * I wonder why User:SCZenz writes: "… define (or 'create' if you will) … ." Chapter 4 of Dedekind's Continuity and Irrational Numbers is entitled, in German, "Schöpfung der irrationalen Zahlen." As translated by W. W. Beman, in the English translation published by Open Court and Dover, this definitely means, "Creation of Irrational Numbers." In this chapter, Dedekind wrote, "… we create a new, an irrational number … ." Beman translated this from the German " … erschaffen wir eine neue, eine irrationale Zahl … ." There is no mention of "define." There is only Dedekind's use of the words meaning creation and creating. We can't arbitrarily change Dedekind's words and freely substitute "define" for "create" whenever we feel like doing so. By adding the phrase, "if you will," User:SCZenz seems to imply that his word "define" is the preferred word, but, the word "create" will be tolerated if necessary. This is not the case. We have Dedekind's own exact words as a reference. He, the original author, used the words meaning creation and creating. If he wanted to use the word meaning "define" he would have written "definieren."Lestrade 18:38, 8 March 2007 (UTC)Lestrade

Dedekind-MacNeille completion
Why does the Dedekind-MacNeille completion consist of the sets (Au)l = A and not of the sets (Al)u = A? Or is this the same? --NeoUrfahraner (talk) 15:35, 23 April 2008 (UTC)
 * In the meanwhile I fonund the answer myself: the sets (Au)l = A are exactly the sets A=Bl, and the sets (Bl)u = B are exactly the sets B=Au, the map $$A\mapsto A^u$$ is order reversing and bijective, its inverse is $$B\mapsto B^l$$, so both give essentially the same completion. --NeoUrfahraner (talk) 08:28, 25 April 2008 (UTC)

Dedekind completions in posets
...One completion of S is the set of its downwardly closed subsets (also called order ideals)...  Downward closed sets are the lower sets; order ideals are directed lower sets; they are not the same. 81.210.248.125 (talk) 05:19, 14 May 2008 (UTC)chrystomath

Anachronism
I have removed the following snippet:
 * The original and most important cases are Dedekind cuts for rational numbers and real numbers. Dedekind used cuts to prove the completeness of the reals without using the axiom of choice (proving the existence of a complete ordered field to be independent of said axiom).

This appears to be an anachronism; Zermelo didn't even formulate AC until 1904, whereas Essays on the Theory of Numbers was published in '01. There may be a valid underlying point here (I suppose if you wanted to prove completeness in terms of Cauchy sequences, you'd be tempted to use AC to pick a representative from each equivalence class of sequences), but it's not really explained. --Trovatore (talk) 06:18, 4 October 2008 (UTC)

sqrt(2) example
I don't understand the example for $$\sqrt(2)$$. It argues that (A,B) is a cut, because if $$x > 0 \wedge x \notin B$$ then there exist $$y > x \wedge y \notin B$$, but how does this prove that A is closed downward, B closed upwards or that A contains no greatest element (like the cut is defined in the introduction)?

Also $$-3 < -1 \in A$$, but $$-3 \notin A$$ and therefore A is not closed downwards, and it shouldn't be a cut. 78.49.23.196 (talk) 12:13, 28 May 2010 (UTC)
 * I have problems with that part, too. Re your last question: similarly to a mistake I just made at Talk:Maximum spacing estimation, you may not have noticed that it says "positive", so your example using negative numbers doesn't apply.  However, the couple of sentences after "neither claim is immediate" need work.  Similar sentences appear in the construction of the real numbers article. Both versions seem not quite right to me.  For one thing, you have to define multiplication of cuts before you can prove things involving multiplication. Both articles have the sentence "However, neither claim is immediate", but seem to be talking about different claims!  I tried to find a ref that would help via Google Books.  This might help a bit:  Elements of the theory of functions By Konrad Knopp. It doesn't do the proof in detail but then neither does this article. ☺ Coppertwig (talk) 20:52, 10 October 2010 (UTC)

 * You, guys are lucky, because it seems you understand something. I don't. What't the meaning of this? Any fool can make an open-bordered set by not including the surface (in Dedekind's case a zero-dimensional surface constituted by the "cut" value), but so what? Wasn't that known before Dedekind? And what's the difference between making a rational or an irrational cut? Anyways the two sets are open towards the cutting value. Rursus dixit. ( m bork3 !) 14:01, 14 October 2010 (UTC)
 * Set A is open: it contains no greatest element. Set B may be open or closed: if it's at a rational number, then it contains a least element, which is that number.  If it's a cut of the rationals and is at an irrational number, then it contains no least element (among the rationals) and is open, like A.
 * OK, any fool can divide a set into two sets. The brilliant thing here is to define the cut as a significant mathematical object and start building a new number system out of it – that is, to start from the rationals and create a model for the set of real numbers. Defining how to multiply cuts is part of that process. ☺ Coppertwig (talk) 16:49, 28 November 2010 (UTC)

Can the cut be rational?
I'm not sure whether the article contradicts itself or is just a bit confusing.

The first paragraph states first that a Dedekind cut is a partition of the rationals into two sets A and B. According to the definition of a partition, this means every rational number is a member of exactly one of A and B. However, it goes on to say "The cut itself is in neither set", thereby contradicting the prior statement in the case where the cut is a rational number.

Of course, if we ignore "The cut itself is in neither set", then we have a clear definition. Equivalently for some $$x \in \mathbb{R}$$, we have $$A = \mathbb{Q} \cap (-\infty, x)$$ and $$B = \mathbb{Q} \cap [x, \infty)$$. Consequently, B has a least element iff x is rational.

I suppose the question is whether "The cut itself is in neither set" means that: — Smjg (talk) 14:21, 13 August 2011 (UTC)
 * the cut is just the notional boundary between the values in A and the values in B, and it merely corresponds to a rational or irrational real number and is not the number itself
 * any cut on a rational number isn't a Dedekind cut (though I'm not sure that the least upper bound property can be satisfied that way)

I agree with your first two statements above. The definition is different from that in "Advanced Calculus" by Angus Taylor. I think the definition is wrong; but I don't know what you mean by Q in your definition of A and B. RHB100 (talk) 21:35, 7 November 2011 (UTC)
 * ℚ is the standard symbol for the set of all rational numbers. See Set (mathematics). — Smjg (talk) 13:08, 15 November 2011 (UTC)

I'm just looking at this and who knows if anyone cares any more, but I'd say that we could avoid the word rational for just a little bit and talk about fractions which are (of course) expressions  $$ a/b $$ with $$a,b$$ relatively prime integers and $$b$$ positive. A (fraction) cut is an ordered pair (A,B) of (infinite) sets of fractions (meeting certain conditions) so it is not a fraction itself. Hence the cut is not in either set since it is not a fraction. Now if A has a largest member $$r=a/b$$ (which is a fraction since all the members are fractions) then that one fraction $$r$$ tells us everything there is to know about the cut (A,B). Now it gets confusing because we define addition and multiplication of cuts. If we call the cuts with a greatest element $$r$$ rationals then these new rationals work just like the fractions and that cut mentioned "is" the rational $$r$$. But the cut is not really the same thing as the largest fraction in the left-hand member of the cut. I hope that helps. Gentlemath (talk) 03:50, 25 April 2013 (UTC)

Relation to Euclidean Geometry needs references/clarification/expansion?
I wonder if the following should provided with references, and the discussion expanded just slightly (although probably in a later section, not in the introduction).

"Dedekind used the German word Schnitt (cut) in a visual sense rooted in Euclidean geometry. When two straight lines cross, one is said to cut the other. Dedekind's construction of the number line ensures that two crossing lines always have one point in common because each of them defines a Dedekind cut on the other."

This suggests a motivation for defining the real numbers, coming from a desire for a two-dimensional space to have the properties one might want, e.g. to satisfy Euclid's postulates. It does seem plausible to me that this was one of Dedekind's motivations, but would like to see a fairly specific reference to establish that this was [an important part of] Dedekind's motivation for the "Schnitt". (I am not familiar with the primary sources so I'm not the person to do this.) The relation of "the number line" (why not say the real numbers) to Euclidean geometry might be made more explicit here, or at least a reference to a rigorous treatment of the proposition claimed ("the construction ensures...") would be very desirable. If that takes up too much space in the introduction, perhaps putting it in a later section would be appropriate. I don't think a detailed treatment is warranted; just something a little bit clearer, e.g. some reference to Dedekind's construction of the reals allowing R^2 to be a model of Euclid's postulates in the two-dimensional case, or something along these lines. If anyone thinks they can do this concisely, and with awareness of the extent to which it reflects Dedekind's motivation, I for one think it would be a valuable improvement to the article.

MorphismOfDoom (talk) 17:45, 18 December 2012 (UTC)

If it seems consistent with the mathematical facts, I think the following wording might improve the passage from the article quoted at the top of my previous comment:

"Dedekind used the German word Schnitt (cut) in a visual sense rooted in Euclidean geometry. When two straight lines cross, one is said to cut the other. Dedekind's construction of the real numbers allows us to use R^2 (the Cartesian product of R with itself) as a model of two-dimensional Euclidean space. In particular, and unlike the Cartesian product of the rationals with themselves, it satisfies Euclid's postulate that two crossing lines always have one point in common because each of them defines a Dedekind cut on the other."

Again I'm not that aware of the history... in some sense the fact that the Cartesian product of the rationals with themselves does not satisfy Euclid's postulates has been known at least since the Greeks, so Descartes was surely aware of it... so it may not be that Dedekind really solved a problem with the Cartesian plane... on the other hand, I guess the point may be that the construction preceded or came simultaneously with the first rigorous axiomatization of the reals? One could argue that, if the rationals were correctly (even if somewhat informally) axiomatized, then Dedekind's construction provides, in some sense, the first rigorous mathematical theory of the real numbers, and therefore of Euclidean two-space, so that it really does solve the problem in a way the Greeks and Descartes, even though they recognized the desirability of talking about irrational numbers, and even did so to a great extent. Anyone care to comment? This seems close to the essence of Dedekind's achievement, so it is central to the article. — Preceding unsigned comment added by MorphismOfDoom (talk • contribs) 17:59, 18 December 2012 (UTC)

Problems with the section on the construction of the reals
The example of the cut representing the square root of 2 is fine. But there's nothing unclear about it actually being a cut. Also, any discussion of it representing the square root of 2 is inappropriate unless some information is added about how the algebraic structure of the rationals is extended to the set of cuts. Otherwise, the discussion trying to show that this really is the square root of 2 doesn't really make any sense. — Preceding unsigned comment added by 73.147.18.72 (talk) 22:36, 13 June 2015 (UTC)
 * The information about the algebraic structure extension should be added anyway, regardless of this issue — the section on constructing the reals is incomplete without it. —David Eppstein (talk) 22:48, 13 June 2015 (UTC)

Transcendentals
The prototypical examples of dedekind cuts are always rational numbers and simple square roots like sqrt(2). I can easily see how dedekind cuts give rise to algebraic numbers using some kind of integer polynomial in the set construction bounding our rationals, such as saying x^2 < 2 as per the case of sqrt(2). I also understand the argument that the "reals" constructed with dedekind cuts have the least upper bound property. What I fail to see, however, is why the "reals" contain the transcendentals. Why cannot the reals complete with the least upper bound property contain only algebraics? That is to say, I dont see how dedekind cuts give rise to transcendentals. I dont see how to express the bounds on the rationals in the lower cut to give rise to, say, pi, for example. It seems to me that constructing the transcendentals with dedekind cuts requires reference to other transcendentals... or best case scenario, non-rational algebraics. I feel when it comes to transcendentals there is a circular reasoning problem going on here. Can someone show me an example of a non-trivial transcendental being defined with a dedekind cut? Or even for that matter a particular algebraic root of a fifth degree polynomial? I have yet to see such a thing in my research. Alternatively, I *can* construct transcendentals if I bound the rationals in the lower cut with an infinite polynomial or infinite series, but then at that point we'd might as well employ the Cauchy construction of the reals instead. 50.35.103.217 (talk) 15:19, 20 September 2018 (UTC)

How can B have the smallest element if A can't be empty?
Since, all elements in set A are smaller than all elements of B and A cant be empty, how could Bhave the smallest element of the rational numbers?

--TheFibonacciEffect (talk) 09:59, 23 April 2020 (UTC)

Addition and multiplication of Dedekind cuts
It would be good to include in this article a discussion of how addition and multiplication of Dedekind cuts are defined. Currently these operations are left unmentioned.

(Addition is pretty easy, but multiplication of cuts can be a bit awkward.)

title: Talk:Dedekind cut


text:
Relic (disambiguation)

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Relic, Relics or The Relic may also refer to:

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Literature

 * Relic (Preston and Child novel), a 1995 novel by Douglas Preston and Lincoln Child
 * The Relic (Queiroz novel), an 1887 novel by José Maria de Eça de Queirós
 * The Relic (Anthony novel), a 1991 novel by Evelyn Anthony
 * Relic, a 2001 novel by Tom Egeland
 * Relic (Foster novel), a 2018 novel by Alan Dean Foster
 * Relic, a comic book enemy of Green Lantern

Music

 * Relics (album), by Pink Floyd, 1971
 * "The Relic", a song by Iced Earth from the 2017 album Incorruptible
 * "The Relic", a song by Symphony X from the 1998 album Twilight in Olympus

title: Relic (disambiguation)


text:
Talk:Absolute infinite

Untitled
I'm not at all happy with this page. I don't know a lot about transfinites but there are things here that worry me. Limit? Not in the normal sense. This needs expansion at least.

You can't just give something a name and hope that it won't introduce inconsistency to do so. Often it does. Going to have a think about this. Andrewa 10:50 16 Jul 2003 (UTC)

This is a proper noun. I have moved the page back to "Absolute Infinite" (with "absolute infinite" redirecting here). -- The Anome 08:55, 18 Sep 2003 (UTC)

In the see also section, what does "The Absolute" refer to?? Jaberwocky6669 02:28, Mar 30, 2005 (UTC)
 * Allow me to redo my question, what is the "absolute" at the top of the article? Jaberwocky6669 02:30, Mar 30, 2005 (UTC)

The sentence " Indeed, naive set theory might be said to be based on this notion. " in Burali-Forti is vague and seems incorrect to me. For example, in the treatment is fully consistent with ZFC. I suggest removing this sentence altogether as it does not add anything useful and confuses the inexpert reader. cerniagigante (talk) 16:34, 24 January 2017 (UTC)

Did he seriously think this?
Is this serious? Did Cantor actually believe that "that every property of the Absolute Infinite is also held by some smaller object"? I mean, he was clearly an incredibly smart guy, but on the face of it, that's an idiotic opinion.

For example, it implies that there is some smaller object that is also larger than all objects besides itself. -Rwv37 04:25, Jun 27, 2005 (UTC)
 * Isn't the universe such an object? 24.174.45.155 21:33, 16 July 2006 (UTC)
 * No. The universe is an object that is larger than all objects besides itself.  But that property is *not* shared with some smaller object. -Rwv37 00:41, 22 July 2006 (UTC)
 * I am using this exchange as justifying this article as "unclear." Cantor's quote "that every property..." is almost certainly a reference to the Reflection Theorem of set theory--but I can't be sure without a citation. See Reflection principle for more information. Cobaltnova 22:14, 10 November 2007 (UTC)

Sense of "multiplicity"
The notion of multiplicity Cantor describes here can't possibly be the same as on multiplicity ''A multiplicity is called well-ordered if it fulfills the condition that every sub-multiplicity has a first element; such a multiplicity I call for short a sequence. '' 24.174.45.155 21:33, 16 July 2006 (UTC)
 * I think he means what we now call a cardinal number. I just piped the wikilink which I hope is ok. 75.62.4.229 (talk) 04:37, 24 November 2007 (UTC)
 * Close. I am virtually certain he just means a set. Because the definition of a well-ordered set is one in which each subset has a least member. Hccrle (talk) 21:36, 29 August 2009 (UTC)

I followed the link and was surprised to arrive at Cardinal number which seemed unhelpful. I therefore added an Other senses section to multiplicity (mathematics) and linked to it. There I said he seemed to mean an ordered set, but it would be helpful if someone with access to the original and translation could check what he meant and provide an exact definition and explanation, or a link to (part of) a different article. PJTraill (talk) 11:31, 6 November 2022 (UTC)
 * My edit there was reverted on the grounds that Wikipedia is not a dictionary and that I was just providing the common-language meaning, but that seems inaccurate. PJTraill (talk) 11:59, 6 November 2022 (UTC)

I started a topic at Talk: Multiplicity (mathematics) to discuss this further —— so doing so here seems redundant! PJTraill (talk) 12:17, 6 November 2022 (UTC)

I have removed the link and added a note linking to set (mathematics). PJTraill (talk) 14:46, 6 November 2022 (UTC)

Absolute Absolute Infinite, Absolute Absolute Absolute Infinite, etc.
The way that leads from Infinite to Absolute Infinite could lead also from Absolute Infinite to Absolute Absolute Infinite, from Absolute Absolute Infinite to Absolute Absolute Absolute Infinite and beyond. So, where is the limit? 89.1.112.168 07:25, 13 September 2007 (UTC)

The limit is at the beginning of your diatribe: the Absolute Infinite is the order of proper classes, which are not sets. Therefore we are not allowed to form bigger sets. So there is nothing larger than the Absolute Infinite. That's what is absolute about it. There is no Absolute Absolute Infinite. Hccrle (talk) 22:00, 29 August 2009 (UTC)

Proper Classes and Philosophical Qualms
I am removing "mysterious" from the description of proper classes. These ideas are mathematically well-defined (see Kunen, Kenneth "Set Theory: An Introduction to Independence Proofs").

von Neumann universe
I think the corresponding idea in axiomatic set theory is the von Neumann universe also known as the cumulative hierarchy. 75.62.4.229 (talk) 04:39, 24 November 2007 (UTC)

You're on the right track. Assuming the Axiom of Choice, the transfinite cardinal numbers are ordinal numbers that form the backbone, so to speak, of the von Neumann universe. Every stage of construction of that universe has a rank, which is an ordinal number and is a member of that stage. The class of all the ordinal numbers is a proper class, and is the Absolute Infinite. Hccrle (talk) 08:19, 30 August 2009 (UTC)

What is bigger than Absolute Infinite?
I shall call him XIBTAI - acronym of Xibtai Is Bigger Than Absolute Infinite! —Preceding unsigned comment added by 89.0.54.122 (talk) 08:27, 11 September 2008 (UTC)

See the above section Absolute Absolute Infinite ... Hccrle (talk) 22:05, 29 August 2009 (UTC)

Nothing is larger than The Absolute infinite
So, what is larger than Nothing? 217.132.68.201 (talk) 11:18, 16 February 2010 (UTC)

Variants of the quote
Currently the article uses this translation:

"The actual infinite arises in three contexts: first when it is realized in the most complete form, in a fully independent otherworldly being, in Deo, where I call it the Absolute Infinite or simply Absolute; second when it occurs in the contingent, created world; third when the mind grasps it in abstracto as a mathematical magnitude, number or order type."

Jané however has this translation in "The role of the absolute infinite in Cantor's conception of set": "The actual infinite can be divided according to three aspects: first, as it is realized in the supreme perfection, in the completely independent, extrawordly being, in God, where I call it absolute infinite or simply absolute; second as it is represented in the dependent world of things created; third as conceived in abstracto as a mathematical quantity, number or ordertype. (Cantor 1887-88, p. 378)"

And from the original quote (https://www.uni-siegen.de/fb6/phima/lehre/phima10/quellentexte/handout-phima-teil4b.pdf) it seems to be:

"The actual infinite was distinguished by three relations: first, as it is realized in the supreme perfection, in the completely independent, extrawordly existence, in Deo, where I call it absolute infinite or simply absolute; second to the extent that it is represented in the dependent, creatural world; third as it can be conceived in abstracto in thought as a mathematical magnitude, number or ordertype. In the latter two relations, where it obviously reveals itself as limited and capable for further proliferation and hence familiar to the finite, I call it Transfinitum and strongly contrast it with the absolute."

--Fixuture (talk) 23:45, 4 July 2015 (UTC)

 * That is a major difference. The first two translations suggest that Cantor was naive; the last seems suggests that Cantor's thinking was quite modern and up-to-date (viz a premonition of 20th century transfinite work). 67.198.37.16 (talk) 04:27, 8 July 2016 (UTC)

 * So I took your word for it and replaced that translation with the 3rd one. Thanks for your review. I included the original quote in German in the reference. --Fixuture (talk) 21:44, 18 July 2016 (UTC)

What's biggest number in world?
it's called Absolutely Absolute Infinite.89.0.54.122 (talk) — Preceding unsigned comment added by 178.90.121.55 (talk) 06:16, 21 August 2023 (UTC)

 * @178.90.121.55 I don't know what you're talking about. do you have evidence? 2601:83:4280:A9E0:282E:701:B11:EFF4 (talk) 19:15, 21 August 2023 (UTC)

This is 10absolute infinite
10absolute infinite — Preceding unsigned comment added by 178.89.254.140 (talk) 12:12, 8 September 2023 (UTC)

 * How 'bout ΩΩ? Guess like zero it's "undefined".  P.I. Ellsworth  , ed.  put'er there 12:27, 13 May 2024 (UTC)

Svg
File:Australia in its region (Ashmore and Cartier Islands special).svg — Preceding unsigned comment added by 95.56.95.231 (talk) 18:57, 10 September 2023 (UTC)

Requested move 1 May 2024

 * The following is a closed discussion of a requested move. Please do not modify it. Subsequent comments should be made in a new section on the talk page. Editors desiring to contest the closing decision should consider a move review after discussing it on the closer's talk page. No further edits should be made to this discussion.

The result of the move request was: moved. To alternative "lowercased infinite". (non-admin closure) Safari Scribe Edits! Talk! 10:05, 13 May 2024 (UTC)

Absolute Infinite → Absolute infinity – according to Infinity. Also, the title of an article must be written in lower case with the initial letter of the remaining words, when they are not proper nouns. --Χιονάκι (talk) 22:16, 30 April 2024 (UTC) — Relisting. ASUKITE  14:58, 9 May 2024 (UTC)

 * Oppose. Sources and other links point to "absolute infinite" as the name of this concept. The article was renamed from absolute infinite in 2003 apparently because an editor thought it to be a "proper name". Sources don't seem to support that, so I suggest that "Infinite" be lowercased and the article should be titled Absolute infinite.  P.I. Ellsworth  , ed.  put'er there 01:34, 9 May 2024 (UTC)
 * Note: WikiProject Logic and WikiProject Philosophy have been notified of this discussion. ASUKITE  14:58, 9 May 2024 (UTC)
 * Relisting comment: Relisting to gather more participation. Please note the alternative of Absolute infinite (lowercase i) proposed above ASUKITE  14:58, 9 May 2024 (UTC)

 * Support lowercase - either infinite or infinity. Primergrey (talk) 16:27, 9 May 2024 (UTC)

 * Support alternative Absolute infinite - per Paine Ellsworth. Psychastes (talk) 17:41, 9 May 2024 (UTC)

The discussion above is closed. Please do not modify it. Subsequent comments should be made on the appropriate discussion page. No further edits should be made to this discussion.

title: Talk:Absolute infinite


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User talk:Will

Greetings! I hope you like the place and decide to stay. If you have questions or doubts of any sort, do not hesitate to post them on the Village Pump, somebody will respond ASAP. Other helpful pages include: Have fun! --Jiang 10:58 16 Jul 2003 (UTC)
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