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Gödel's incompleteness theorems

Gödel's incompleteness theorems are two theorems of mathematical logic that demonstrate the inherent limitations of every formal axiomatic system capable of modelling basic arithmetic. These results, published by Kurt Gödel in 1931, are important both in mathematical logic and in the philosophy of mathematics; the theorems are but not universally, interpreted as showing that Hilbert's program to find a complete and consistent set of axioms for all mathematics is impossible. The first incompleteness theorem states that no consistent system of axioms whose theorems can be listed by an effective procedure is capable of proving all truths about the arithmetic of natural numbers. For any such consistent formal system, there will always be statements about natural numbers that are true, but that are unprovable within the system; the second incompleteness theorem, an extension of the first, shows that the system cannot demonstrate its own consistency. Employing a diagonal argument, Gödel's incompleteness theorems were the first of several related theorems on the limitations of formal systems.

They were followed by Tarski's undefinability theorem on the formal undefinability of truth, Church's proof that Hilbert's Entscheidungsproblem is unsolvable, Turing's theorem that there is no algorithm to solve the halting problem. The incompleteness theorems apply to formal systems that are of sufficient complexity to express the basic arithmetic of the natural numbers and which are consistent, axiomatized, these concepts being detailed below. In the context of first-order logic, formal systems are called formal theories. In general, a formal system is a deductive apparatus that consists of a particular set of axioms along with rules of symbolic manipulation that allow for the derivation of new theorems from the axioms. One example of such a system is first-order Peano arithmetic, a system in which all variables are intended to denote natural numbers. In other systems, such as set theory, only some sentences of the formal system express statements about the natural numbers; the incompleteness theorems are about formal provability within these systems, rather than about "provability" in an informal sense.

There are several properties that a formal system may have, including completeness and the existence of an effective axiomatization. The incompleteness theorems show that systems which contain a sufficient amount of arithmetic cannot possess all three of these properties. A formal system is said to be axiomatized if its set of theorems is a recursively enumerable set; this means that there is a computer program that, in principle, could enumerate all the theorems of the system without listing any statements that are not theorems. Examples of generated theories include Peano arithmetic and Zermelo–Fraenkel set theory; the theory known as true arithmetic consists of all true statements about the standard integers in the language of Peano arithmetic. This theory is consistent, complete, contains a sufficient amount of arithmetic; however it does not have a recursively enumerable set of axioms, thus does not satisfy the hypotheses of the incompleteness theorems. A set of axioms is complete if, for any statement in the axioms' language, that statement or its negation is provable from the axioms.

This is the notion relevant for Gödel's first Incompleteness theorem. It is not to be confused with semantic completeness, which means that the set of axioms proves all the semantic tautologies of the given language. In his completeness theorem, Gödel proved, but it is not syntactically complete, since there are sentences expressible in the language of first order logic that can be neither proved nor disproved from the axioms of logic alone. In a mere system of logic it would be absurd to expect syntactic completeness, but in a system of mathematics, thinkers such as Hilbert had believed that it is just a matter of time to find such an axiomatization that would allow one to either prove or disprove each and every mathematical formula. A formal system might be syntactically incomplete by design, as logics are. Or it may be incomplete because not all the necessary axioms have been discovered or included. For example, Euclidean geometry without the parallel postulate is incomplete, because some statements in the language can not be proved from the remaining axioms.

The theory of dense linear orders is not complete, but becomes complete with an extra axiom stating that there are no endpoints in the order. The continuum hypothesis is a statement in the language of ZFC, not provable within ZFC, so ZFC is not complete. In this case, there is no obvious candidate for a new axiom; the theory of first order Peano arithmetic seems to be consistent. Assuming this is indeed the case, note that it has an infinite but recursively enumerable set of axioms, can encode enough arithmetic for the hypotheses of the incompleteness theorem, thus by the first incompleteness theorem, Peano Arithmetic is not complete. The theorem gives an explicit example of a statement of arithmetic, neither provable nor disprovable in Peano's arithmetic. Moreover, this statement is true in the usual model. In addition, no axiomatized, consistent extension of Peano arithmetic can be complete. A set of axioms is consistent if there is no statement such that both the statement and its negation are provable from the axioms, inconsistent otherwise.

Peano arithmetic is provably consistent from

Jean de Montereul

Jean de Montreuil was a French ecclesiastic and diplomat. The son of an advocate to the parlement de Paris, Montereul was intended for a legal career himself, but in the course of a trip to Italy with Pomponne de Bellièvre he was made a canon of Toul. Paul Pellisson wrote that he was "very proper in negotiation, with a flexible and agile mind concerted, who would hardly do anything without a purpose, it was he who gave the opinion that the Elector Palatine would have to pass into France incognito, to go to command the Duke of Weimar's troops, seize de Brissac. In November 1637 when Bellièvre was appointed ambassador to the Court of St James, Montereul joined him at Charles I's court accepting the post of secretary to the ambassador, he stayed in London as chargé d'affaires as when Bellièvre left in late January 1640, returning to France himself in early summer 1641 when the Marquis de La Ferté-Imbault was appointed ambassador. For just over two years from February 1642 until the spring of 1644 Montereul was secretary to the Marquis de Fontenay-Mareuil, the French ambassador in Rome.

When it looked that the Royalists would loose the English Civil War, the Scots who were allied with the English Parliamentarians, looked to Cardinal Mazarin, by the chief minister of France for help in securing Charles I's position as king, but on terms acceptable to the Scots. In response Mazarin appointed Montereul as French resident in Scotland. Montereul arrived in London in August 1645. Once there he opened a dialogue with English Presbyterians such as the Earl of Holland who were sympathetic to the Scots who too were Presbyterians and formally allied with the Roundheads thorough the Solemn League and Covenant, but, disliked by non-Presbyterian Roundheads such as Oliver Cromwell and other religious Independents. There were Scottish commissioners in London who were looking after Scottish interests in the alliance and during talks with them and English Presbyterians, the idea arose that if Charles I was to place himself under the protection of the Scottish army the Presbyterian party could advance their interests.

In January 1646, in furtherance of this plan Montereul journeyed to the Royals headquarters in Oxford where he met Charles I. However Charles was at first unwilling to accept Montereul's proposals as he was unhappy with the proposed abolition of the episcopacy in both kingdoms. Montereul returned to London empty handed. Back in London there was nearly a diplomatic incident when Montereul on discovering that the Algernon Percy, Earl of Northumberland, had obtained French as yet unopened diplomatic correspondence, retrieved them, causing Percy to considered detaining the Montereul by force, but in the end the Earl backed down. In March Montereul returned to Oxford, without the agreement of Scots, but in the name of the French king, promised Charles that he would be received by the Scots as their rightful sovereign. Having made that promise Montereul journeyed north and the Scottish Army encampment on outskirts of Newark-on-Trent, he lodged at King's Arms Inn in Southwell.. After further correspondence and with few options open to him, Charles I journeyed to the Scottish camp and placed himself under the protection of the Scottish commander David Leslie, Lord Newark.

It became apparent to Montereul that what the Scots would do and what Charles I was willing to give were not as close as Montereul had led both parties to expect. In June Montereul returned to France with Charles holding out false hopes of French diplomatic aid. Cardinal Mazarin decide not follow Montereul's advice and decided to send Bellièvre to England as an ambassador. Montereul returned to Newark, but on the way is diplomatic dispatches were ceased and read by Parliamentary officers; when their contents became public Charles's credibility suffered, because what they contained showed that his private commitments differed from those he was publicly putting forwards. When Charles was transferred to English custody and lodged in Holmby House, Montereul left him and travelled to Edinburgh. Here he maintained good relations with both the Engagers and the party that would become known as the Whigamores. Since Montereul's first arrival in England a key concern of Mazarin had been the importance of raising troops for French service.

Rendering assistance to Charles was for the cardinal a secondary concern, doubly so once it became that the English Parliament would emerge victorious. The numbers of soldiers that Montereul was able to raise remained low, so from Mazarin's perspective Montereul efforts England was not fruitful. On his return to France in July 1647, Montreuil resumed his role as secretary to the prince of Conti, went to Rome again in 1648 on his return to Paris, elected to the Académie française in 1649. Remaining faithful to the prince of Conti and to the duke of Longueville, he entered into a secret correspondence with them during their imprisonment in 1650, but died of tuberculosis aged 36 or 37, shortly after they were freed. Besides a few pieces of verse and prose which were not published, he left an abundant correspondence, published in Edinburgh at the end of the 19th century; the Diplomatic correspondence of Jean de Montereul and the brothers de Bellièvre, French ambassadors in England and Scotland, with an English translation and notes, by J. G. Fotheringham, 2 volumes, Édimbourg, 1898–1899 Bienassis, Loic.

"Montreuil, Jean de". Oxford Dictionary of National Biography. Oxford University Press. Doi:10.1093/ref:odnb/105837. (S

Flunked

Flunked is a 2008 documentary film conceived by and executive produced by Steven Maggi, directed by Corey Burres and narrated by actor Joe Mantegna. It explores problems in the United States public education system and reviews successful education reform solutions in both charter and public schools, letting leading educators tell their stories. Flunked studies the poor position of the United States public education system; the film though, explores some of the system's successes. The first 20 minutes review many of the system's problems, as well as schools nationwide that prepared students well for college in the 2000s. Based on their high test scores, their graduates seemed capable of working and competing in tomorrow's economy; the documentary shows ways to reform troubled public schools, as well as alternatives to them, including charter schools. Jim Anderson Steve Barr Ben Chavis Bill Cosby Andrew Coulson Eric Dominguez Heidi Dominguez Angie Dorman Bill Gates Donn Harris Lynn Harsh Guilbert Hentschke Bob Herbold Charlie Hoff Therese Holliday Karen Jones Jeff Kropf Howard LappinSteven Maggi Joe Mantegna, narrator John Merrifield Dan Nicklay Dennis Pantano Bill Proser David Scortt Jason Singer Caitlyn Snaring Traci Snaring Matt Sween Matt Wingard Flunked won Best Documentary at the San Fernando Valley International Film Festival in Los Angeles, Best Educational Documentary at the Bayou City Inspirational Film Festival in Houston, the Award of Merit from the Accolade Competition, the first SPNovation Award.

Official website Flunked on IMDb

Batavia Institute

The Batavia Institute is a Registered Historic Place in Batavia, Illinois, US. Batavia Institute, a private academy, was chartered on 12 February 1853 by 13 men, including Rev. Stephen Peet, the Congregational minister, Elijah Shumway Town, Joel McKee, John Van Nortwick, Dennison K. Town, who settled in Batavia in 1839 as its first physician, Isaac G. Wilson; the building's central part, which still stands in Batavia at 333 South Jefferson Street, at Union Avenue, was constructed in 1853–1854 of locally quarried limestone at a cost of $20,000. The architect Elijah Shumway Town designed the building in a Greek Revival style. At the time the Batavia Institute was built, there were no secondary schools in Batavia. In fact, since not many towns had high schools, students came to the Batavia Institute from all over Illinois; the school operated for over 10 years under the supervision of the area's Congregational churches until new public school laws lessened the need for such a school. For a short time the building was rented to the public schools.

Using the Batavia Institute as the basis for its proposal, Batavia submitted a bid for the Illinois normal school in 1857. A normal school or teachers college is an educational institution for training teachers, its purpose is to establish teaching norms, hence its name. The State of Illinois passed an act to establish a normal school on 18 February 1857—the second west of the Appalachian Mountains. Bids were opened by the State Board of Education in Peoria on 7 May 1857; the first proposition on the agenda was from Batavia, which offered a subscription of $15,000, with the land and building belonging to the Batavia Institute, valued at $30,000, making $45,000 in all. Washington, in Tazewell County and Peoria submitted proposals, as well. After considerable discussion, a resolution was adopted locating the new university at Bloomington—actually north of town at the village of North Bloomington, renamed Normal in 1865, for the school. Illinois State University celebrated its 150th anniversary in 2007.

The building and grounds of the Batavia Institute were sold in 1867 to Dr. Richard J. Patterson, who, as proprietor and medical superintendent, operated it as a private rest home and sanitarium for women, called Bellevue Place; the sanitarium operated until July 1965. The most notable patient was Mary Todd Lincoln, widow of U. S. President Abraham Lincoln, a patient for several months in the summer of 1875. In the 1960s, the building was converted to a residential facility for unwed mothers called the Fox Hill Home; the Fox Hill Home operated into the 1970s. In the middle of the 1980s, the building was once again named Bellevue Place and converted into apartments; the building was listed in the National Register of Historic Place on 13 August 1976

Vampire Princess Miyu

Vampire Princess Miyu is a Japanese horror manga series by Narumi Kakinouchi and Toshiki Hirano, as well as an anime adaptation by the same creators. The anime was presented in a 4-episode OVA licensed by AnimEigo in 1988, was adapted into a 26-episode television series licensed by Tokyopop and released in 1997. Stranded in the space between the human world and the demon underworld, the series central characters are a Japanese vampire girl named Miyu and her Western Shinma companion Larva. Miyu is the daughter of both a Shinma, she was born a vampire and as such, she was awakened as the guardian whose destiny is to hunt down all stray Shinma and send them back to "the Darkness". Before turning 15 years old, she yearns to return to the darkness herself but not until she has banished all the Shinma from Earth, and since her awakening, she remains what she is. Most locations in the series are evocative of traditional Japan. Miyu Miyu Voiced by: Naoko Watanabe, Pamela Wiedner, Anne Marie Zola Miyu Voiced by: Miki Nagasawa, Kimberly J. Brown, Dorothy Elias-Fahn A beautiful girl who appears to be around 13 or 15 years old but in fact is much older, being a vampire.

In Japanese, "Miyu" means "beauty of the evening", "beautiful evening" or "evening beauty". She can teleport, open dimensional portals, use fire attacks; when in her vampire appearance, Miyu is always barefoot in the snow as she doesn't get cold. In the OVA, she is a vampire mother in post -- World War II Japan. In the TV series, on the other hand, her mother is human and her father is a Shinma guardian in pre-World War II Empire of Japan. In both cases, Miyu becomes the Guardian after losing her parents. In the OVA, Miyu is depicted as child-like and playful, flamboyant in talking when having a conversation with Himiko, while the TV series' Miyu is more reserved and composed. Though she is a vampire, Miyu is not harmed by sunlight, holy water or crucifixes, her reflection can be seen; this is because she is technically a Daywalker having one vampire. She needs to drink blood to survive and she chooses her'victims' since she cannot take blood from others unless they give it to her willingly. So, Miyu picks people whom she believes to be "lovely" who have suffered a tragic loss, offers them their greatest wish – to be with their lost loved ones, at least in their dreams – in exchange for their blood.

These people live in an endless dream state. Miyu has great concern for him. In the TV series, when posing as a human, she goes by the name Miyu Yamano. In the OVA, she is seen to wear different clothes in every episode. For example in the first OVA, she wears her typical short kimono and light purple obi, familiarized by all fans where her right foot has a ribbon-like wrapping around it. In the second OVA, she wears a Japanese school uniform during her stay at school and wears a bright red yukata when speaking to Himiko and confronting Ranka. In the third OVA, she wears a winter kimono which seems to cover her body more than the other two garments. In the fourth OVA, she wears a heavy black kimono and wears a mask; however in the TV series, she only wears two types of clothing: the typical Japanese school uniform and the kimono known by all fans and, shown on box covers. Larva Larva Voiced by: Kaneto Shiozawa. In the OVA, Larva comes to prevent Miyu's vampire blood from awakening and kill her, but he inadvertently triggers it and she drinks his blood when he drops his guard.

As a result of this failure, Larva's face and voice are sealed behind a mask for all eternity. In the TV series, Larva faces Miyu. After he has struck her down, she drinks his blood. In both cases, Larva starts out as an unwilling ally, but pledges to be by Miyu's side because he can sense her sorrow, something he was able to glimpse at during both blood-bond scenes. Larva can use his nails to slash things and he can wield a scythe, he is powerful, as he is seen to win against every shinma without getting mortally wounded. In addition to this, in the TV series and the manga, he is able to access Miyu's flame powers, but does so as it reminds him of his greatest defeat. Unlike in the OVA, in the TV series and Manga, Larva can speak and will remove his mask; the name "Larva" is taken from Roman mythology, which like "Lemures" refers to a restless spirit of the dead. Many of the Western Shinma appearing in the second manga series follow a naming convention drawn from various European demons and spirits.

Larva's name and signature mask may have been inspired from the white, ghostly Venetian Carnival mask "Larva," called "Volto." In Volume 1 of the TV series, Larva was cal

Tonight for Sure

Tonight for Sure is a 1962 Western softcore comedy film directed by Francis Ford Coppola. It was written by Jerry Shaffer. Jack Hill was the Director of Photography; the music was composed by Carmine Coppola. It is a film set in August 1961 on the Sunset Strip starring Karl Schanzer and Don Kenney and featuring Electra, Laura Cornell, Karla Lee, Sue Martin; the film features footage from The Peeper, an unfinished Western set in a nudist colony. On the Sunset Strip, two unlikely men rendezvous: Samuel Hill, an unkempt desert miner, Benjamin Jabowski, a John Birch Society dandy from the city. Intent on some sort of mayhem, they enter the Herald Club before the burlesque show starts, they wire something to the electrical box, set to blow at midnight, they sit at the back of the club to get to know each other. As they drink and glance at the stage, Sam tells of a partner driven mad by visions of naked women in the sagebrush; as they get drunker and the clock ticks toward midnight, they pull their chairs closer to the women on stage.

List of American films of 1962 The Party at Kitty and Stud's, the debut of Sylvester Stallone Sugar Cookies, a film produced by Oliver Stone Caligula, with Malcolm McDowell, Peter O'Toole, John Gielgud and Helen Mirren Abel Ferrara, former pornographic film director Jerry Stahl, former pornographic screenwriter Tonight for Sure on IMDb Tonight for Sure at the TCM Movie Database Tonight for Sure at AllMovie