title: Mary Pickford

text: Her appeal was summed up two years later by the February 1916 issue of "Photoplay" as "luminous tenderness in a steel band of gutter ferocity". Only Charlie Chaplin, who slightly surpassed Pickford's popularity in 1916, had a similarly spellbinding pull with critics and the audience. Each enjoyed a level of fame far exceeding that of other actors. Throughout the 1910s and 1920s, Pickford was believed to be the most famous woman in the world, or, as a silent-film journalist described her, "the best known woman who has ever lived, the woman who was known to more people and loved by more people than any other woman that has been in all history".


title: Mary Pickford

text: Pickford starred in 52 features throughout her career. On June 24, 1916, Pickford signed a new contract with Zukor that granted her full authority over production of the films in which she starred, and a record-breaking salary of $10,000 a week. In addition, Pickford's compensation was half of a film's profits, with a guarantee of $1,040,000 (US$ in 2022), making her the first actress to sign a million dollar contract. She also became vice-president of Pickford Film Corporation.


title: Mary Pickford

text: Occasionally, she played a child, in films such as "The Poor Little Rich Girl" (1917), "Rebecca of Sunnybrook Farm" (1917), "Daddy-Long-Legs" (1919), and "Pollyanna" (1920). Pickford's fans were devoted to these "little girl" roles, but they were not typical of her career. Due to her lack of a normal childhood, she enjoyed making these pictures. Given how small she was at under five feet, and her naturalistic acting abilities, she was very successful in these roles. Douglas Fairbanks Jr., when he first met her in person as a boy, assumed she was a new playmate for him, and asked her to come and play trains with him, which she obligingly did.


title: Mary Pickford

text: In August 1918, Pickford's contract expired and, when refusing Zukor's terms for a renewal, she was offered $250,000 to leave the motion picture business. She declined, and went to First National Pictures, which agreed to her terms. In 1919, Pickford, along with D. W. Griffith, Charlie Chaplin, and Douglas Fairbanks, formed the independent film production company United Artists. Through United Artists, Pickford continued to produce and perform in her own movies; she could also distribute them as she chose. In 1920, Pickford's film "Pollyanna" grossed around $1,100,000. The following year, Pickford's film "Little Lord Fauntleroy" was also a success, and in 1923, "Rosita" grossed over $1,000,000 as well. During this period, she also made "Little Annie Rooney" (1925), another film in which Pickford played a child, "Sparrows" (1926), which blended the Dickensian with newly minted German expressionist style, and "My Best Girl" (1927), a romantic comedy featuring her future husband Charles "Buddy" Rogers.


title: Mary Pickford

text: The arrival of sound was her undoing. Pickford underestimated the value of adding sound to movies, claiming that "adding sound to movies would be like putting lipstick on the Venus de Milo".


title: Mary Pickford

text: She played a reckless socialite in "Coquette" (1929), her first talkie, a role for which her famous ringlets were cut into a 1920s' bob. Pickford had already cut her hair in the wake of her mother's death in 1928. Fans were shocked at the transformation. Pickford's hair had become a symbol of female virtue, and when she cut it, the act made front-page news in "The New York Times" and other papers. "Coquette" was a success and won her an Academy Award for Best Actress, although this was highly controversial. The public failed to respond to her in the more sophisticated roles. Like most movie stars of the silent era, Pickford found her career fading as talkies became more popular among audiences.


title: Mary Pickford

text: Her next film, "The Taming of The Shrew", made with husband Douglas Fairbanks, was not well received at the box office. Established Hollywood actors were panicked by the impending arrival of the talkies. On March 29, 1928, "The Dodge Brothers Hour" was broadcast from Pickford's bungalow, featuring Fairbanks, Chaplin, Norma Talmadge, Gloria Swanson, John Barrymore, D. W. Griffith, and Dolores del Río, among others. They spoke on the radio show to prove that they could meet the challenge of talking movies.


title: Mary Pickford

text: A transition in the roles Pickford selected came when she was in her late 30s, no longer able to play the children, teenage spitfires, and feisty young women so adored by her fans, and not suited for the glamorous and vampish heroines of early sound. In 1933, she underwent a Technicolor screen test for an animated/live-action film version of "Alice in Wonderland", but Walt Disney discarded the project when Paramount released its own version of the book. Only one Technicolor still of her screen test still exists.


title: Mary Pickford

text: She retired from film acting in 1933 following three costly failures with her last film appearance being "Secrets". She appeared on stage in Chicago in 1934 in the play "The Church Mouse" and went on tour in 1935, starting in Seattle with the stage version of "Coquette". She also appeared in a season of radio plays for NBC in 1935 and CBS in 1936. In 1936 she became vice-president of United Artists and continued to produce films for others, including "One Rainy Afternoon" (1936), "The Gay Desperado" (1936), "Sleep, My Love" (1948; with Claudette Colbert), and "Love Happy" (1949), with the Marx Brothers.


title: Mary Pickford

text: Pickford used her stature in the movie industry to promote a variety of causes. Although her image depicted fragility and innocence, she proved to be a strong businesswoman who took control of her career in a cutthroat industry.


title: Mary Pickford

text: During World War I she promoted the sale of Liberty Bonds, making an intensive series of fund-raising speeches, beginning in Washington, D.C., where she sold bonds alongside Charlie Chaplin, Douglas Fairbanks, Theda Bara, and Marie Dressler. Five days later she spoke on Wall Street to an estimated 50,000 people. Though Canadian-born, she was a powerful symbol of American culture, kissing the American flag for cameras and auctioning one of her world-famous curls for $15,000. In a single speech in Chicago, she sold an estimated five million dollars' worth of bonds. She was christened the U.S. Navy's official "Little Sister"; the Army named two cannons after her and made her an honorary colonel.


title: Mary Pickford

text: In 1916, Pickford and Constance Adams DeMille, wife of director Cecil B. DeMille, helped found the Hollywood Studio Club, a dormitory for young women involved in the motion picture business. At the end of World War I, Pickford conceived of the Motion Picture Relief Fund, an organization to help financially needy actors. Leftover funds from her work selling Liberty Bonds were put toward its creation, and in 1921, the Motion Picture Relief Fund (MPRF) was officially incorporated, with Joseph Schenck voted its first president and Pickford its vice president. In 1932, Pickford spearheaded the "Payroll Pledge Program", a payroll-deduction plan for studio workers who gave one half of one percent of their earnings to the MPRF. As a result, in 1940, the Fund was able to purchase land and build the Motion Picture Country House and Hospital, in Woodland Hills, California.


title: Mary Pickford

text: An astute businesswoman, Pickford became her own producer within three years of her start in features. According to her Foundation, "she oversaw every aspect of the making of her films, from hiring talent and crew to overseeing the script, the shooting, the editing, to the final release and promotion of each project". She demanded (and received) these powers in 1916, when she was under contract to Zukor's Famous Players in Famous Plays (later Paramount). Zukor acquiesced to her refusal to participate in block-booking, the widespread practice of forcing an exhibitor to show a bad film of the studio's choosing to also be able to show a Pickford film. In 1916, Pickford's films were distributed, singly, through a special distribution unit called Artcraft. The Mary Pickford Corporation was briefly Pickford's motion-picture production company.


title: Mary Pickford

text: In 1919, she increased her power by co-founding United Artists (UA) with Charlie Chaplin, D. W. Griffith, and her soon-to-be husband, Douglas Fairbanks. Before UA's creation, Hollywood studios were vertically integrated, not only producing films but forming chains of theaters. Distributors (also part of the studios) arranged for company productions to be shown in the company's movie venues. Filmmakers relied on the studios for bookings; in return they put up with what many considered creative interference.


title: Mary Pickford

text: United Artists broke from this tradition. It was solely a distribution company, offering independent film producers access to its own screens as well as the rental of temporarily unbooked cinemas owned by other companies. Pickford and Fairbanks produced and shot their films after 1920 at the jointly owned Pickford-Fairbanks studio on Santa Monica Boulevard. The producers who signed with UA were true independents, producing, creating and controlling their work to an unprecedented degree. As a co-founder, as well as the producer and star of her own films, Pickford became the most powerful woman who has ever worked in Hollywood. By 1930, Pickford's acting career had largely faded. After retiring three years later, however, she continued to produce films for United Artists. She and Chaplin remained partners in the company for decades. Chaplin left the company in 1955, and Pickford followed suit in 1956, selling her remaining shares for $3 million.


title: Mary Pickford

text: She had bought the rights to many of her early silent films with the intention of burning them on her death, but in 1970 she agreed to donate 50 of her Biograph films to the American Film Institute. In 1976, she received an Academy Honorary Award for her contribution to American film.


title: Mary Pickford

text: Pickford was married three times. She married Owen Moore, an Irish-born silent film actor, on January 7, 1911. It is rumored she became pregnant by Moore in the early 1910s and had a miscarriage or an abortion. Some accounts suggest this resulted in her later inability to have children. The couple's marriage was strained by Moore's alcoholism, insecurity about living in the shadow of Pickford's fame, and bouts of domestic violence. The couple lived together on-and-off for several years.


title: Mary Pickford

text: Pickford became secretly involved in a relationship with Douglas Fairbanks. They toured the U.S. together in 1918 to promote Liberty Bond sales for the World War I effort. Around this time, Pickford also suffered from the flu during the 1918 flu pandemic. Pickford divorced Moore on March 2, 1920, after she agreed to his $100,000 demand for a settlement ($1.4 million in 2021, adjusted for inflation). She married Fairbanks just days later on March 28, 1920, in what was described as the "marriage of the century" and they were referred to as the King and Queen of Hollywood. They went to Europe for their honeymoon; fans in London and in Paris caused riots trying to get to the famous couple. The couple's triumphant return to Hollywood was witnessed by vast crowds who turned out to hail them at railway stations across the United States.


title: Mary Pickford

text: "The Mark of Zorro" (1920) and a series of other swashbucklers gave the popular Fairbanks a more romantic, heroic image. Pickford continued to epitomize the virtuous but fiery girl next door. Even at private parties, people instinctively stood up when Pickford entered a room; she and her husband were often referred to as "Hollywood royalty". Their international reputations were broad. Foreign heads of state and dignitaries who visited the White House often asked if they could also visit Pickfair, the couple's mansion in Beverly Hills.


title: Mary Pickford

text: Dinners at Pickfair became celebrity events. Charlie Chaplin, Fairbanks' best friend, was often present. Other guests included George Bernard Shaw, Albert Einstein, Elinor Glyn, Helen Keller, H. G. Wells, Lord Mountbatten, Fritz Kreisler, Amelia Earhart, F. Scott Fitzgerald, Noël Coward, Max Reinhardt, Baron Nishi, Vladimir Nemirovich-Danchenko, Sir Arthur Conan Doyle, Austen Chamberlain, Sir Harry Lauder, and Meher Baba, among others. However, the public nature of Pickford's second marriage strained it to the breaking point. Both she and Fairbanks had little time off from producing and acting in their films. They were also constantly on display as America's unofficial ambassadors to the world, leading parades, cutting ribbons, and making speeches. When their film careers both began to flounder at the end of the silent era, Fairbanks' restless nature prompted him to overseas travel (something which Pickford did not enjoy). When Fairbanks' romance with Sylvia, Lady Ashley became public in the early 1930s, he and Pickford separated. They divorced January 10, 1936. Fairbanks' son by his first wife, Douglas Fairbanks Jr., claimed his father and Pickford long regretted their inability to reconcile.


title: Mary Pickford

text: On June 24, 1937, Pickford married her third and last husband, actor and band leader Charles "Buddy" Rogers. They adopted two children: Roxanne (born 1944, adopted 1944) and Ronald Charles (born 1937, adopted 1943, a.k.a. Ronnie Pickford Rogers). A PBS "American Experience" documentary described Pickford's relationship with her children as tense. She criticized their physical imperfections, including Ronnie's small stature and Roxanne's crooked teeth. Both children later said their mother was too self-absorbed to provide real maternal love. In 2003, Ronnie recalled that "Things didn't work out that much, you know. But I'll never forget her. I think that she was a good woman."


title: Mary Pickford

text: Pickford supported Thomas Dewey in the 1944 United States presidential election, Barry Goldwater in the 1964 United States presidential election and Ronald Reagan in his race for governor in 1966.


title: Mary Pickford

text: After retiring from the screen, Pickford became an alcoholic, as her father had been. Her mother Charlotte died of breast cancer in March 1928. Her siblings, Lottie and Jack, both died of alcohol-related causes in 1936 and 1933, respectively. These deaths, her divorce from Fairbanks, and the end of silent films left Pickford deeply depressed. Her relationship with her adopted children, Roxanne and Ronald, was turbulent at best. Pickford withdrew and gradually became a recluse, remaining almost entirely at Pickfair and allowing visits only from Lillian Gish, her stepson Douglas Fairbanks Jr., and a few select others.


title: Mary Pickford

text: In 1955, she published her memoirs, "Sunshine and Shadows". She had previously published "Why Not Try God" in 1934, an essay on spirituality and personal growth, "My Rendevouz of Life" (1935), an essay on death and her belief in an afterlife and also a novel in 1935, "The Demi-Widow". She appeared in court in 1959, in a matter pertaining to her co-ownership of North Carolina TV station WSJS-TV. The court date coincided with the date of her 67th birthday; under oath, when asked to give her age, Pickford replied: "I'm 21, going on 20."


title: Mary Pickford

text: In the mid-1960s, Pickford often received visitors only by telephone, speaking to them from her bedroom. Charles "Buddy" Rogers often gave guests tours of Pickfair, including views of a genuine western bar Pickford had bought for Douglas Fairbanks, and a portrait of Pickford in the drawing room. A print of this image now hangs in the Library of Congress. When Pickford received an Academy Honorary Award in 1976, the Academy sent a TV crew to her house to record her short statement of thanks – offering the public a very rare glimpse into Pickfair Manor. Charitable events continued to be held at Pickfair, including an annual Christmas party for blind war veterans, mostly from World War I.


title: Mary Pickford

text: Pickford believed that she had ceased to be a British subject when she married Fairbanks, an American citizen, in 1920. Thus, she never acquired Canadian citizenship when it was first created in 1947. However, Pickford held and traveled under a British/Canadian passport which she renewed regularly at the British/Canadian consulates in Los Angeles, and she did not take out papers for American citizenship. She also owned a house in Toronto, Ontario, Canada. Toward the end of her life, Pickford made arrangements with the Canadian Department of Citizenship to officially acquire Canadian citizenship because she wished to "die as a Canadian". Canadian authorities were not sure that she had ever lost her Canadian citizenship, given her passport status, but her request was approved and she officially became a Canadian citizen.


title: Mary Pickford

text: On May 29, 1979, Pickford died at a Santa Monica, California, hospital of complications from a cerebral hemorrhage she had suffered the week before. She was interred in the Garden of Memory of the Forest Lawn Memorial Park cemetery in Glendale, California.


title: Gödel's incompleteness theorems

text: Gödel's incompleteness theorems are two theorems of mathematical logic that are concerned with the limits of in formal axiomatic theories. These results, published by Kurt Gödel in 1931, are important both in mathematical logic and in the philosophy of mathematics. The theorems are widely, but not universally, interpreted as showing that Hilbert's program to find a complete and consistent set of axioms for all mathematics is impossible.


title: Gödel's incompleteness theorems

text: The first incompleteness theorem states that no consistent system of axioms whose theorems can be listed by an effective procedure (i.e., an algorithm) 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.


title: Gödel's incompleteness theorems

text: Employing a diagonal argument, Gödel's incompleteness theorems were the first of several closely 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, and Turing's theorem that there is no algorithm to solve the halting problem.


title: Gödel's incompleteness theorems

text: 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 and effectively axiomatized. Particularly in the context of first-order logic, formal systems are also 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 (or rules of inference) 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.


title: Gödel's incompleteness theorems

text: There are several properties that a formal system may have, including completeness, consistency, 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.


title: Gödel's incompleteness theorems

text: A formal system is said to be "effectively axiomatized" (also called "effectively generated") if its set of theorems is a recursively enumerable set .


title: Gödel's incompleteness theorems

text: 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 effectively generated theories include Peano arithmetic and Zermelo–Fraenkel set theory (ZFC).


title: Gödel's incompleteness theorems

text: 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 and complete, and contains a sufficient amount of arithmetic. However it does not have a recursively enumerable set of axioms, and thus does not satisfy the hypotheses of the incompleteness theorems.


title: Gödel's incompleteness theorems

text: A set of axioms is ("syntactically", or "negation"-) 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 (not to be confused with the incompleteness theorems described here), Gödel proved that first order logic is "semantically" complete. 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.


title: Gödel's incompleteness theorems

text: 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 (by proving its negation) each and every mathematical formula.


title: Gödel's incompleteness theorems

text: A formal system might be syntactically incomplete by design, as logics generally are. Or it may be incomplete simply 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 (such as the parallel postulate itself) can not be proved from the remaining axioms. Similarly, 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 that is not provable within ZFC, so ZFC is not complete. In this case, there is no obvious candidate for a new axiom that resolves the issue.


title: Gödel's incompleteness theorems

text: 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, and 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 that is neither provable nor disprovable in Peano's arithmetic. Moreover, this statement is true in the usual model. In addition, no effectively axiomatized, consistent extension of Peano arithmetic can be complete.


title: Gödel's incompleteness theorems

text: A set of axioms is (simply) consistent if there is no statement such that both the statement and its negation are provable from the axioms, and "inconsistent" otherwise. That is to say, a consistent axiomatic system is one that is free from contradiction.


title: Gödel's incompleteness theorems

text: Peano arithmetic is provably consistent from ZFC, but not from within itself. Similarly, ZFC is not provably consistent from within itself, but ZFC + "there exists an inaccessible cardinal" proves ZFC is consistent because if is the least such cardinal, then sitting inside the von Neumann universe is a model of ZFC, and a theory is consistent if and only if it has a model.


title: Gödel's incompleteness theorems

text: If one takes all statements in the language of Peano arithmetic as axioms, then this theory is complete, has a recursively enumerable set of axioms, and can describe addition and multiplication. However, it is not consistent.


title: Gödel's incompleteness theorems

text: Additional examples of inconsistent theories arise from the paradoxes that result when the axiom schema of unrestricted comprehension is assumed in set theory.


title: Gödel's incompleteness theorems

text: The incompleteness theorems apply only to formal systems which are able to prove a sufficient collection of facts about the natural numbers. One sufficient collection is the set of theorems of Robinson arithmetic . Some systems, such as Peano arithmetic, can directly express statements about natural numbers. Others, such as ZFC set theory, are able to interpret statements about natural numbers into their language. Either of these options is appropriate for the incompleteness theorems.


title: Gödel's incompleteness theorems

text: The theory of algebraically closed fields of a given characteristic is complete, consistent, and has an infinite but recursively enumerable set of axioms. However it is not possible to encode the integers into this theory, and the theory cannot describe arithmetic of integers. A similar example is the theory of real closed fields, which is essentially equivalent to Tarski's axioms for Euclidean geometry. So Euclidean geometry itself (in Tarski's formulation) is an example of a complete, consistent, effectively axiomatized theory.


title: Gödel's incompleteness theorems

text: The system of Presburger arithmetic consists of a set of axioms for the natural numbers with just the addition operation (multiplication is omitted). Presburger arithmetic is complete, consistent, and recursively enumerable and can encode addition but not multiplication of natural numbers, showing that for Gödel's theorems one needs the theory to encode not just addition but also multiplication.


title: Gödel's incompleteness theorems

text: In choosing a set of axioms, one goal is to be able to prove as many correct results as possible, without proving any incorrect results. For example, we could imagine a set of true axioms which allow us to prove every true arithmetical claim about the natural numbers . In the standard system of first-order logic, an inconsistent set of axioms will prove every statement in its language (this is sometimes called the principle of explosion), and is thus automatically complete. A set of axioms that is both complete and consistent, however, proves a maximal set of non-contradictory theorems .


title: Gödel's incompleteness theorems

text: The pattern illustrated in the previous sections with Peano arithmetic, ZFC, and ZFC + "there exists an inaccessible cardinal" cannot generally be broken. Here ZFC + "there exists an inaccessible cardinal" cannot from itself, be proved consistent. It is also not complete, as illustrated by the continuum hypothesis, which is unresolvable in ZFC + "there exists an inaccessible cardinal".


title: Gödel's incompleteness theorems

text: The first incompleteness theorem shows that, in formal systems that can express basic arithmetic, a complete and consistent finite list of axioms can never be created: each time an additional, consistent statement is added as an axiom, there are other true statements that still cannot be proved, even with the new axiom. If an axiom is ever added that makes the system complete, it does so at the cost of making the system inconsistent. It is not even possible for an infinite list of axioms to be complete, consistent, and effectively axiomatized.


title: Gödel's incompleteness theorems

text: Gödel's first incompleteness theorem first appeared as "Theorem VI" in Gödel's 1931 paper "On Formally Undecidable Propositions of Principia Mathematica and Related Systems I". The hypotheses of the theorem were improved shortly thereafter by using Rosser's trick. The resulting theorem (incorporating Rosser's improvement) may be paraphrased in English as follows, where "formal system" includes the assumption that the system is effectively generated.


title: Gödel's incompleteness theorems

text: First Incompleteness Theorem: "Any consistent formal system within which a certain amount of elementary arithmetic can be carried out is incomplete; i.e., there are statements of the language of which can neither be proved nor disproved in ." (Raatikainen 2020)


title: Gödel's incompleteness theorems

text: The unprovable statement referred to by the theorem is often referred to as "the Gödel sentence" for the system . The proof constructs a particular Gödel sentence for the system , but there are infinitely many statements in the language of the system that share the same properties, such as the conjunction of the Gödel sentence and any logically valid sentence.


title: Gödel's incompleteness theorems

text: Each effectively generated system has its own Gödel sentence. It is possible to define a larger system  that contains the whole of plus as an additional axiom. This will not result in a complete system, because Gödel's theorem will also apply to , and thus also cannot be complete. In this case, is indeed a theorem in , because it is an axiom. Because states only that it is not provable in , no contradiction is presented by its provability within . However, because the incompleteness theorem applies to , there will be a new Gödel statement for , showing that is also incomplete. will differ from in that will refer to , rather than .


title: Gödel's incompleteness theorems

text: The Gödel sentence is designed to refer, indirectly, to itself. The sentence states that, when a particular sequence of steps is used to construct another sentence, that constructed sentence will not be provable in . However, the sequence of steps is such that the constructed sentence turns out to be itself. In this way, the Gödel sentence indirectly states its own unprovability within .


title: Gödel's incompleteness theorems

text: To prove the first incompleteness theorem, Gödel demonstrated that the notion of provability within a system could be expressed purely in terms of arithmetical functions that operate on Gödel numbers of sentences of the system. Therefore, the system, which can prove certain facts about numbers, can also indirectly prove facts about its own statements, provided that it is effectively generated. Questions about the provability of statements within the system are represented as questions about the arithmetical properties of numbers themselves, which would be decidable by the system if it were complete.


title: Gödel's incompleteness theorems

text: Thus, although the Gödel sentence refers indirectly to sentences of the system , when read as an arithmetical statement the Gödel sentence directly refers only to natural numbers. It asserts that no natural number has a particular property, where that property is given by a primitive recursive relation . As such, the Gödel sentence can be written in the language of arithmetic with a simple syntactic form. In particular, it can be expressed as a formula in the language of arithmetic consisting of a number of leading universal quantifiers followed by a quantifier-free body (these formulas are at level formula_1 of the arithmetical hierarchy). Via the MRDP theorem, the Gödel sentence can be re-written as a statement that a particular polynomial in many variables with integer coefficients never takes the value zero when integers are substituted for its variables .


title: Gödel's incompleteness theorems

text: The first incompleteness theorem shows that the Gödel sentence of an appropriate formal theory is unprovable in . Because, when interpreted as a statement about arithmetic, this unprovability is exactly what the sentence (indirectly) asserts, the Gödel sentence is, in fact, true (; also see ). For this reason, the sentence is often said to be "true but unprovable." . However, since the Gödel sentence cannot itself formally specify its intended interpretation, the truth of the sentence may only be arrived at via a meta-analysis from outside the system. In general, this meta-analysis can be carried out within the weak formal system known as primitive recursive arithmetic, which proves the implication , where is a canonical sentence asserting the consistency of (, ).


title: Gödel's incompleteness theorems

text: Although the Gödel sentence of a consistent theory is true as a statement about the intended interpretation of arithmetic, the Gödel sentence will be false in some nonstandard models of arithmetic, as a consequence of Gödel's completeness theorem . That theorem shows that, when a sentence is independent of a theory, the theory will have models in which the sentence is true and models in which the sentence is false. As described earlier, the Gödel sentence of a system is an arithmetical statement which claims that no number exists with a particular property. The incompleteness theorem shows that this claim will be independent of the system , and the truth of the Gödel sentence follows from the fact that no standard natural number has the property in question. Any model in which the Gödel sentence is false must contain some element which satisfies the property within that model. Such a model must be "nonstandard" – it must contain elements that do not correspond to any standard natural number (, ).


title: Gödel's incompleteness theorems

text: Gödel specifically cites Richard's paradox and the liar paradox as semantical analogues to his syntactical incompleteness result in the introductory section of "On Formally Undecidable Propositions in Principia Mathematica and Related Systems I". The liar paradox is the sentence "This sentence is false." An analysis of the liar sentence shows that it cannot be true (for then, as it asserts, it is false), nor can it be false (for then, it is true). A Gödel sentence for a system makes a similar assertion to the liar sentence, but with truth replaced by provability: says " is not provable in the system ." The analysis of the truth and provability of is a formalized version of the analysis of the truth of the liar sentence.


title: Gödel's incompleteness theorems

text: It is not possible to replace "not provable" with "false" in a Gödel sentence because the predicate " is the Gödel number of a false formula" cannot be represented as a formula of arithmetic. This result, known as Tarski's undefinability theorem, was discovered independently both by Gödel, when he was working on the proof of the incompleteness theorem, and by the theorem's namesake, Alfred Tarski.


title: Gödel's incompleteness theorems

text: Compared to the theorems stated in Gödel's 1931 paper, many contemporary statements of the incompleteness theorems are more general in two ways. These generalized statements are phrased to apply to a broader class of systems, and they are phrased to incorporate weaker consistency assumptions.


title: Gödel's incompleteness theorems

text: Gödel demonstrated the incompleteness of the system of "Principia Mathematica", a particular system of arithmetic, but a parallel demonstration could be given for any effective system of a certain expressiveness. Gödel commented on this fact in the introduction to his paper, but restricted the proof to one system for concreteness. In modern statements of the theorem, it is common to state the effectiveness and expressiveness conditions as hypotheses for the incompleteness theorem, so that it is not limited to any particular formal system. The terminology used to state these conditions was not yet developed in 1931 when Gödel published his results.


title: Gödel's incompleteness theorems

text: Gödel's original statement and proof of the incompleteness theorem requires the assumption that the system is not just consistent but "ω-consistent". A system is ω-consistent if it is not ω-inconsistent, and is ω-inconsistent if there is a predicate such that for every specific natural number the system proves , and yet the system also proves that there exists a natural number such that (). That is, the system says that a number with property exists while denying that it has any specific value. The ω-consistency of a system implies its consistency, but consistency does not imply ω-consistency. strengthened the incompleteness theorem by finding a variation of the proof (Rosser's trick) that only requires the system to be consistent, rather than ω-consistent. This is mostly of technical interest, because all true formal theories of arithmetic (theories whose axioms are all true statements about natural numbers) are ω-consistent, and thus Gödel's theorem as originally stated applies to them. The stronger version of the incompleteness theorem that only assumes consistency, rather than ω-consistency, is now commonly known as Gödel's incompleteness theorem and as the Gödel–Rosser theorem.


title: Gödel's incompleteness theorems

text: For each formal system containing basic arithmetic, it is possible to canonically define a formula Cons() expressing the consistency of . This formula expresses the property that "there does not exist a natural number coding a formal derivation within the system whose conclusion is a syntactic contradiction." The syntactic contradiction is often taken to be "0=1", in which case Cons() states "there is no natural number that codes a derivation of '0=1' from the axioms of ."


title: Gödel's incompleteness theorems

text: Gödel's second incompleteness theorem shows that, under general assumptions, this canonical consistency statement Cons() will not be provable in . The theorem first appeared as "Theorem XI" in Gödel's 1931 paper "On Formally Undecidable Propositions in Principia Mathematica and Related Systems I". In the following statement, the term "formalized system" also includes an assumption that is effectively axiomatized.


title: Gödel's incompleteness theorems

text: Second Incompleteness Theorem: "For any consistent system "F" within which a certain amount of elementary arithmetic can be carried out, the consistency of "F" cannot be proved in "F" itself."


title: Gödel's incompleteness theorems

text: "Assume is a consistent formalized system which contains elementary arithmetic. Then formula_2." (Then "F" does not prove consistency of "F")


title: Gödel's incompleteness theorems

text: This theorem is stronger than the first incompleteness theorem because the statement constructed in the first incompleteness theorem does not directly express the consistency of the system. The proof of the second incompleteness theorem is obtained by formalizing the proof of the first incompleteness theorem within the system itself.


title: Gödel's incompleteness theorems

text: There is a technical subtlety in the second incompleteness theorem regarding the method of expressing the consistency of as a formula in the language of . There are many ways to express the consistency of a system, and not all of them lead to the same result. The formula Cons() from the second incompleteness theorem is a particular expression of consistency.


title: Gödel's incompleteness theorems

text: Other formalizations of the claim that is consistent may be inequivalent in , and some may even be provable. For example, first-order Peano arithmetic (PA) can prove that "the largest consistent subset of PA" is consistent. But, because PA is consistent, the largest consistent subset of PA is just PA, so in this sense PA "proves that it is consistent". What PA does not prove is that the largest consistent subset of PA is, in fact, the whole of PA. (The term "largest consistent subset of PA" is meant here to be the largest consistent initial segment of the axioms of PA under some particular effective enumeration.)


title: Gödel's incompleteness theorems

text: The standard proof of the second incompleteness theorem assumes that the provability predicate satisfies the Hilbert–Bernays provability conditions. Letting represent the Gödel number of a formula , the provability conditions say:


title: Gödel's incompleteness theorems

text: There are systems, such as Robinson arithmetic, which are strong enough to meet the assumptions of the first incompleteness theorem, but which do not prove the Hilbert–Bernays conditions. Peano arithmetic, however, is strong enough to verify these conditions, as are all theories stronger than Peano arithmetic.


title: Gödel's incompleteness theorems

text: Gödel's second incompleteness theorem also implies that a system satisfying the technical conditions outlined above cannot prove the consistency of any system that proves the consistency of . This is because such a system can prove that if proves the consistency of , then is in fact consistent. For the claim that is consistent has form "for all numbers , has the decidable property of not being a code for a proof of contradiction in ". If were in fact inconsistent, then would prove for some that is the code of a contradiction in . But if also proved that is consistent (that is, that there is no such ), then it would itself be inconsistent. This reasoning can be formalized in to show that if is consistent, then is consistent. Since, by second incompleteness theorem, does not prove its consistency, it cannot prove the consistency of either.


title: Gödel's incompleteness theorems

text: This corollary of the second incompleteness theorem shows that there is no hope of proving, for example, the consistency of Peano arithmetic using any finitistic means that can be formalized in a system the consistency of which is provable in Peano arithmetic (PA). For example, the system of primitive recursive arithmetic (PRA), which is widely accepted as an accurate formalization of finitistic mathematics, is provably consistent in PA. Thus PRA cannot prove the consistency of PA. This fact is generally seen to imply that Hilbert's program, which aimed to justify the use of "ideal" (infinitistic) mathematical principles in the proofs of "real" (finitistic) mathematical statements by giving a finitistic proof that the ideal principles are consistent, cannot be carried out .


title: Gödel's incompleteness theorems

text: The corollary also indicates the epistemological relevance of the second incompleteness theorem. It would actually provide no interesting information if a system proved its consistency. This is because inconsistent theories prove everything, including their consistency. Thus a consistency proof of in would give us no clue as to whether really is consistent; no doubts about the consistency of would be resolved by such a consistency proof. The interest in consistency proofs lies in the possibility of proving the consistency of a system in some system that is in some sense less doubtful than itself, for example weaker than . For many naturally occurring theories and , such as = Zermelo–Fraenkel set theory and = primitive recursive arithmetic, the consistency of is provable in , and thus cannot prove the consistency of by the above corollary of the second incompleteness theorem.


title: Gödel's incompleteness theorems

text: The second incompleteness theorem does not rule out altogether the possibility of proving the consistency of some theory , only doing so in a theory that itself can prove to be consistent. For example, Gerhard Gentzen proved the consistency of Peano arithmetic in a different system that includes an axiom asserting that the ordinal called is wellfounded; see Gentzen's consistency proof. Gentzen's theorem spurred the development of ordinal analysis in proof theory.


title: Gödel's incompleteness theorems

text: There are two distinct senses of the word "undecidable" in mathematics and computer science. The first of these is the proof-theoretic sense used in relation to Gödel's theorems, that of a statement being neither provable nor refutable in a specified deductive system. The second sense, which will not be discussed here, is used in relation to computability theory and applies not to statements but to decision problems, which are countably infinite sets of questions each requiring a yes or no answer. Such a problem is said to be undecidable if there is no computable function that correctly answers every question in the problem set (see undecidable problem).


title: Gödel's incompleteness theorems

text: Because of the two meanings of the word undecidable, the term independent is sometimes used instead of undecidable for the "neither provable nor refutable" sense.


title: Gödel's incompleteness theorems

text: Undecidability of a statement in a particular deductive system does not, in and of itself, address the question of whether the truth value of the statement is well-defined, or whether it can be determined by other means. Undecidability only implies that the particular deductive system being considered does not prove the truth or falsity of the statement. Whether there exist so-called "absolutely undecidable" statements, whose truth value can never be known or is ill-specified, is a controversial point in the philosophy of mathematics.


title: Gödel's incompleteness theorems

text: The combined work of Gödel and Paul Cohen has given two concrete examples of undecidable statements (in the first sense of the term): The continuum hypothesis can neither be proved nor refuted in ZFC (the standard axiomatization of set theory), and the axiom of choice can neither be proved nor refuted in ZF (which is all the ZFC axioms "except" the axiom of choice). These results do not require the incompleteness theorem. Gödel proved in 1940 that neither of these statements could be disproved in ZF or ZFC set theory. In the 1960s, Cohen proved that neither is provable from ZF, and the continuum hypothesis cannot be proved from ZFC.


title: Gödel's incompleteness theorems

text: In 1973, Saharon Shelah showed that the Whitehead problem in group theory is undecidable, in the first sense of the term, in standard set theory.


title: Gödel's incompleteness theorems

text: Gregory Chaitin produced undecidable statements in algorithmic information theory and proved another incompleteness theorem in that setting. Chaitin's incompleteness theorem states that for any system that can represent enough arithmetic, there is an upper bound such that no specific number can be proved in that system to have Kolmogorov complexity greater than . While Gödel's theorem is related to the liar paradox, Chaitin's result is related to Berry's paradox.


title: Gödel's incompleteness theorems

text: These are natural mathematical equivalents of the Gödel "true but undecidable" sentence. They can be proved in a larger system which is generally accepted as a valid form of reasoning, but are undecidable in a more limited system such as Peano Arithmetic.


title: Gödel's incompleteness theorems

text: In 1977, Paris and Harrington proved that the Paris–Harrington principle, a version of the infinite Ramsey theorem, is undecidable in (first-order) Peano arithmetic, but can be proved in the stronger system of second-order arithmetic. Kirby and Paris later showed that Goodstein's theorem, a statement about sequences of natural numbers somewhat simpler than the Paris–Harrington principle, is also undecidable in Peano arithmetic.


title: Gödel's incompleteness theorems

text: Kruskal's tree theorem, which has applications in computer science, is also undecidable from Peano arithmetic but provable in set theory. In fact Kruskal's tree theorem (or its finite form) is undecidable in a much stronger system codifying the principles acceptable based on a philosophy of mathematics called predicativism. The related but more general graph minor theorem (2003) has consequences for computational complexity theory.


title: Gödel's incompleteness theorems

text: The incompleteness theorem is closely related to several results about undecidable sets in recursion theory.


title: Gödel's incompleteness theorems

text: Chaitin's incompleteness theorem gives a different method of producing independent sentences, based on Kolmogorov complexity. Like the proof presented by Kleene that was mentioned above, Chaitin's theorem only applies to theories with the additional property that all their axioms are true in the standard model of the natural numbers. Gödel's incompleteness theorem is distinguished by its applicability to consistent theories that nonetheless include statements that are false in the standard model; these theories are known as ω-inconsistent.


title: Gödel's incompleteness theorems

text: The proof by contradiction has three essential parts. To begin, choose a formal system that meets the proposed criteria:


title: Gödel's incompleteness theorems

text: The main problem in fleshing out the proof described above is that it seems at first that to construct a statement that is equivalent to " cannot be proved", would somehow have to contain a reference to , which could easily give rise to an infinite regress. Gödel's technique is to show that statements can be matched with numbers (often called the arithmetization of syntax) in such a way that ""proving a statement"" can be replaced with ""testing whether a number has a given property"". This allows a self-referential formula to be constructed in a way that avoids any infinite regress of definitions. The same technique was later used by Alan Turing in his work on the "Entscheidungsproblem".


title: Gödel's incompleteness theorems

text: In simple terms, a method can be devised so that every formula or statement that can be formulated in the system gets a unique number, called its Gödel number, in such a way that it is possible to mechanically convert back and forth between formulas and Gödel numbers. The numbers involved might be very long indeed (in terms of number of digits), but this is not a barrier; all that matters is that such numbers can be constructed. A simple example is how English can be stored as a sequence of numbers for each letter and then combined into a single larger number:


title: Gödel's incompleteness theorems

text: In principle, proving a statement true or false can be shown to be equivalent to proving that the number matching the statement does or doesn't have a given property. Because the formal system is strong enough to support reasoning about "numbers in general", it can support reasoning about "numbers that represent formulae and statements" as well. Crucially, because the system can support reasoning about "properties of numbers", the results are equivalent to reasoning about "provability of their equivalent statements".


title: Gödel's incompleteness theorems

text: Having shown that in principle the system can indirectly make statements about provability, by analyzing properties of those numbers representing statements it is now possible to show how to create a statement that actually does this.


title: Gödel's incompleteness theorems

text: A formula that contains exactly one free variable is called a "statement form" or "class-sign". As soon as is replaced by a specific number, the statement form turns into a "bona fide" statement, and it is then either provable in the system, or not. For certain formulas one can show that for every natural number , is true if and only if it can be proved (the precise requirement in the original proof is weaker, but for the proof sketch this will suffice). In particular, this is true for every specific arithmetic operation between a finite number of natural numbers, such as "23 = 6".


title: Gödel's incompleteness theorems

text: Statement forms themselves are not statements and therefore cannot be proved or disproved. But every statement form can be assigned a Gödel number denoted by . The choice of the free variable used in the form () is not relevant to the assignment of the Gödel number .


title: Gödel's incompleteness theorems

text: The notion of provability itself can also be encoded by Gödel numbers, in the following way: since a proof is a list of statements which obey certain rules, the Gödel number of a proof can be defined. Now, for every statement , one may ask whether a number is the Gödel number of its proof. The relation between the Gödel number of and , the potential Gödel number of its proof, is an arithmetical relation between two numbers. Therefore, there is a statement form that uses this arithmetical relation to state that a Gödel number of a proof of exists:


title: Gödel's incompleteness theorems

text: The name Bew is short for "beweisbar", the German word for "provable"; this name was originally used by Gödel to denote the provability formula just described. Note that "" is merely an abbreviation that represents a particular, very long, formula in the original language of ; the string "" itself is not claimed to be part of this language.


title: Gödel's incompleteness theorems

text: An important feature of the formula is that if a statement is provable in the system then is also provable. This is because any proof of would have a corresponding Gödel number, the existence of which causes to be satisfied.


title: Gödel's incompleteness theorems

text: The next step in the proof is to obtain a statement which, indirectly, asserts its own unprovability. Although Gödel constructed this statement directly, the existence of at least one such statement follows from the diagonal lemma, which says that for any sufficiently strong formal system and any statement form there is a statement such that the system proves


title: Gödel's incompleteness theorems

text: and the defined by this roughly states that its own Gödel number is the Gödel number of an unprovable formula.


title: Gödel's incompleteness theorems

text: The statement is not literally equal to ; rather, states that if a certain calculation is performed, the resulting Gödel number will be that of an unprovable statement. But when this calculation is performed, the resulting Gödel number turns out to be the Gödel number of itself. This is similar to the following sentence in English:


title: Gödel's incompleteness theorems

text: This sentence does not directly refer to itself, but when the stated transformation is made the original sentence is obtained as a result, and thus this sentence indirectly asserts its own unprovability. The proof of the diagonal lemma employs a similar method.


title: Gödel's incompleteness theorems

text: Now, assume that the axiomatic system is ω-consistent, and let be the statement obtained in the previous section.


title: Gödel's incompleteness theorems

text: If were provable, then would be provable, as argued above. But asserts the negation of . Thus the system would be inconsistent, proving both a statement and its negation. This contradiction shows that cannot be provable.


title: Gödel's incompleteness theorems

text: If the negation of were provable, then would be provable (because was constructed to be equivalent to the negation of ). However, for each specific number , cannot be the Gödel number of the proof of , because is not provable (from the previous paragraph). Thus on one hand the system proves there is a number with a certain property (that it is the Gödel number of the proof of ), but on the other hand, for every specific number , we can prove that it does not have this property. This is impossible in an ω-consistent system. Thus the negation of is not provable.


title: Gödel's incompleteness theorems

text: Thus the statement is undecidable in our axiomatic system: it can neither be proved nor disproved within the system.


title: Gödel's incompleteness theorems

text: In fact, to show that is not provable only requires the assumption that the system is consistent. The stronger assumption of ω-consistency is required to show that the negation of is not provable. Thus, if is constructed for a particular system:


title: Gödel's incompleteness theorems

text: If one tries to "add the missing axioms" to avoid the incompleteness of the system, then one has to add either or "not " as axioms. But then the definition of "being a Gödel number of a proof" of a statement changes. which means that the formula is now different. Thus when we apply the diagonal lemma to this new Bew, we obtain a new statement , different from the previous one, which will be undecidable in the new system if it is ω-consistent.


title: Gödel's incompleteness theorems

text: sketches an alternative proof of the first incompleteness theorem that uses Berry's paradox rather than the liar paradox to construct a true but unprovable formula. A similar proof method was independently discovered by Saul Kripke . Boolos's proof proceeds by constructing, for any computably enumerable set of true sentences of arithmetic, another sentence which is true but not contained in . This gives the first incompleteness theorem as a corollary. According to Boolos, this proof is interesting because it provides a "different sort of reason" for the incompleteness of effective, consistent theories of arithmetic .


title: Gödel's incompleteness theorems

text: The incompleteness theorems are among a relatively small number of nontrivial theorems that have been transformed into formalized theorems that can be completely verified by proof assistant software. Gödel's original proofs of the incompleteness theorems, like most mathematical proofs, were written in natural language intended for human readers.


title: Gödel's incompleteness theorems

text: Computer-verified proofs of versions of the first incompleteness theorem were announced by Natarajan Shankar in 1986 using Nqthm , by Russell O'Connor in 2003 using Coq and by John Harrison in 2009 using HOL Light . A computer-verified proof of both incompleteness theorems was announced by Lawrence Paulson in 2013 using Isabelle .


title: Gödel's incompleteness theorems

text: The main difficulty in proving the second incompleteness theorem is to show that various facts about provability used in the proof of the first incompleteness theorem can be formalized within a system using a formal predicate for provability. Once this is done, the second incompleteness theorem follows by formalizing the entire proof of the first incompleteness theorem within the system itself.


title: Gödel's incompleteness theorems

text: Let stand for the undecidable sentence constructed above, and assume for purposes of obtaining a contradiction that the consistency of the system can be proved from within the system itself. This is equivalent to proving the statement "System is consistent".


title: Gödel's incompleteness theorems

text: Now consider the statement , where = "If the system is consistent, then is not provable". The proof of sentence can be formalized within the system , and therefore the statement , " is not provable", (or identically, "not ") can be proved in the system .


title: Gödel's incompleteness theorems

text: Observe then, that if we can prove that the system is consistent (ie. the statement in the hypothesis of ), then we have proved that is not provable. But this is a contradiction since by the 1st Incompleteness Theorem, this sentence (ie. what is implied in the sentence , """ is not provable") is what we construct to be unprovable. Notice that this is why we require formalizing the first Incompleteness Theorem in : to prove the 2nd Incompleteness Theorem, we obtain a contradiction with the 1st Incompleteness Theorem which can do only by showing that the theorem holds in . So we cannot prove that the system is consistent. And the 2nd Incompleteness Theorem statement follows.


title: Gödel's incompleteness theorems

text: The incompleteness results affect the philosophy of mathematics, particularly versions of formalism, which use a single system of formal logic to define their principles.


title: Gödel's incompleteness theorems

text: The incompleteness theorem is sometimes thought to have severe consequences for the program of logicism proposed by Gottlob Frege and Bertrand Russell, which aimed to define the natural numbers in terms of logic . Bob Hale and Crispin Wright argue that it is not a problem for logicism because the incompleteness theorems apply equally to first order logic as they do to arithmetic. They argue that only those who believe that the natural numbers are to be defined in terms of first order logic have this problem.


title: Gödel's incompleteness theorems

text: Many logicians believe that Gödel's incompleteness theorems struck a fatal blow to David Hilbert's second problem, which asked for a finitary consistency proof for mathematics. The second incompleteness theorem, in particular, is often viewed as making the problem impossible. Not all mathematicians agree with this analysis, however, and the status of Hilbert's second problem is not yet decided (see "Modern viewpoints on the status of the problem").


title: Gödel's incompleteness theorems

text: Authors including the philosopher J. R. Lucas and physicist Roger Penrose have debated what, if anything, Gödel's incompleteness theorems imply about human intelligence. Much of the debate centers on whether the human mind is equivalent to a Turing machine, or by the Church–Turing thesis, any finite machine at all. If it is, and if the machine is consistent, then Gödel's incompleteness theorems would apply to it.


title: Gödel's incompleteness theorems

text: Douglas Hofstadter, in his books "Gödel, Escher, Bach" and "I Am a Strange Loop", cites Gödel's theorems as an example of what he calls a "strange loop", a hierarchical, self-referential structure existing within an axiomatic formal system. He argues that this is the same kind of structure which gives rise to consciousness, the sense of "I", in the human mind. While the self-reference in Gödel's theorem comes from the Gödel sentence asserting its own unprovability within the formal system of Principia Mathematica, the self-reference in the human mind comes from the way in which the brain abstracts and categorises stimuli into "symbols", or groups of neurons which respond to concepts, in what is effectively also a formal system, eventually giving rise to symbols modelling the concept of the very entity doing the perception.


title: Gödel's incompleteness theorems

text: Hofstadter argues that a strange loop in a sufficiently complex formal system can give rise to a "downward" or "upside-down" causality, a situation in which the normal hierarchy of cause-and-effect is flipped upside-down. In the case of Gödel's theorem, this manifests, in short, as the following:


title: Gödel's incompleteness theorems

text: "Merely from knowing the formula's meaning, one can infer its truth or falsity without any effort to derive it in the old-fashioned way, which requires one to trudge methodically "upwards" from the axioms. This is not just peculiar; it is astonishing. Normally, one cannot merely look at what a mathematical conjecture says and simply appeal to the content of that statement on its own to deduce whether the statement is true or false." ("I Am a Strange Loop.")


title: Gödel's incompleteness theorems

text: In the case of the mind, a far more complex formal system, this "downward causality" manifests, in Hofstadter's view, as the ineffable human instinct that the causality of our minds lies on the high level of desires, concepts, personalities, thoughts and ideas, rather than on the low level of interactions between neurons or even fundamental particles, even though according to physics the latter seems to possess the causal power.


title: Gödel's incompleteness theorems

text: "There is thus a curious upside-downness to our normal human way of perceiving the world: we are built to perceive “big stuff” rather than “small stuff”, even though the domain of the tiny seems to be where the actual motors driving reality reside." ("I Am a Strange Loop.")


title: Gödel's incompleteness theorems

text: Although Gödel's theorems are usually studied in the context of classical logic, they also have a role in the study of paraconsistent logic and of inherently contradictory statements ("dialetheia"). argues that replacing the notion of formal proof in Gödel's theorem with the usual notion of informal proof can be used to show that naive mathematics is inconsistent, and uses this as evidence for dialetheism. The cause of this inconsistency is the inclusion of a truth predicate for a system within the language of the system . gives a more mixed appraisal of the applications of Gödel's theorems to dialetheism.


title: Gödel's incompleteness theorems

text: Appeals and analogies are sometimes made to the incompleteness theorems in support of arguments that go beyond mathematics and logic. Several authors have commented negatively on such extensions and interpretations, including Torkel Franzén (2005); Panu ; ; and . , for example, quote from Rebecca Goldstein's comments on the disparity between Gödel's avowed Platonism and the anti-realist uses to which his ideas are sometimes put. criticize Régis Debray's invocation of the theorem in the context of sociology; Debray has defended this use as metaphorical (ibid.).


title: Gödel's incompleteness theorems

text: After Gödel published his proof of the completeness theorem as his doctoral thesis in 1929, he turned to a second problem for his habilitation. His original goal was to obtain a positive solution to Hilbert's second problem . At the time, theories of the natural numbers and real numbers similar to second-order arithmetic were known as "analysis", while theories of the natural numbers alone were known as "arithmetic".


title: Gödel's incompleteness theorems

text: Gödel was not the only person working on the consistency problem. Ackermann had published a flawed consistency proof for analysis in 1925, in which he attempted to use the method of ε-substitution originally developed by Hilbert. Later that year, von Neumann was able to correct the proof for a system of arithmetic without any axioms of induction. By 1928, Ackermann had communicated a modified proof to Bernays; this modified proof led Hilbert to announce his belief in 1929 that the consistency of arithmetic had been demonstrated and that a consistency proof of analysis would likely soon follow. After the publication of the incompleteness theorems showed that Ackermann's modified proof must be erroneous, von Neumann produced a concrete example showing that its main technique was unsound (; ).


title: Gödel's incompleteness theorems

text: In the course of his research, Gödel discovered that although a sentence which asserts its own falsehood leads to paradox, a sentence that asserts its own non-provability does not. In particular, Gödel was aware of the result now called Tarski's indefinability theorem, although he never published it. Gödel announced his first incompleteness theorem to Carnap, Feigel and Waismann on August 26, 1930; all four would attend the Second Conference on the Epistemology of the Exact Sciences, a key conference in Königsberg the following week.


title: Gödel's incompleteness theorems

text: The 1930 Königsberg conference was a joint meeting of three academic societies, with many of the key logicians of the time in attendance. Carnap, Heyting, and von Neumann delivered one-hour addresses on the mathematical philosophies of logicism, intuitionism, and formalism, respectively . The conference also included Hilbert's retirement address, as he was leaving his position at the University of Göttingen. Hilbert used the speech to argue his belief that all mathematical problems can be solved. He ended his address by saying,


title: Gödel's incompleteness theorems

text: This speech quickly became known as a summary of Hilbert's beliefs on mathematics (its final six words, ""Wir müssen wissen. Wir werden wissen!"", were used as Hilbert's epitaph in 1943). Although Gödel was likely in attendance for Hilbert's address, the two never met face to face .


title: Gödel's incompleteness theorems

text: Gödel announced his first incompleteness theorem at a roundtable discussion session on the third day of the conference. The announcement drew little attention apart from that of von Neumann, who pulled Gödel aside for conversation. Later that year, working independently with knowledge of the first incompleteness theorem, von Neumann obtained a proof of the second incompleteness theorem, which he announced to Gödel in a letter dated November 20, 1930 . Gödel had independently obtained the second incompleteness theorem and included it in his submitted manuscript, which was received by "Monatshefte für Mathematik" on November 17, 1930.


title: Gödel's incompleteness theorems

text: Gödel's paper was published in the "Monatshefte" in 1931 under the title "Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I" ("On Formally Undecidable Propositions in Principia Mathematica and Related Systems I"). As the title implies, Gödel originally planned to publish a second part of the paper in the next volume of the "Monatshefte"; the prompt acceptance of the first paper was one reason he changed his plans .


title: Gödel's incompleteness theorems

text: Gödel gave a series of lectures on his theorems at Princeton in 1933–1934 to an audience that included Church, Kleene, and Rosser. By this time, Gödel had grasped that the key property his theorems required is that the system must be effective (at the time, the term "general recursive" was used). Rosser proved in 1936 that the hypothesis of ω-consistency, which was an integral part of Gödel's original proof, could be replaced by simple consistency, if the Gödel sentence was changed in an appropriate way. These developments left the incompleteness theorems in essentially their modern form.


title: Gödel's incompleteness theorems

text: Gentzen published his consistency proof for first-order arithmetic in 1936. Hilbert accepted this proof as "finitary" although (as Gödel's theorem had already shown) it cannot be formalized within the system of arithmetic that is being proved consistent.


title: Gödel's incompleteness theorems

text: The impact of the incompleteness theorems on Hilbert's program was quickly realized. Bernays included a full proof of the incompleteness theorems in the second volume of "Grundlagen der Mathematik" (1939), along with additional results of Ackermann on the ε-substitution method and Gentzen's consistency proof of arithmetic. This was the first full published proof of the second incompleteness theorem.


title: Gödel's incompleteness theorems

text: used a version of Richard's paradox to construct an expression that was false but unprovable in a particular, informal framework he had developed. Gödel was unaware of this paper when he proved the incompleteness theorems (Collected Works Vol. IV., p. 9). Finsler wrote to Gödel in 1931 to inform him about this paper, which Finsler felt had priority for an incompleteness theorem. Finsler's methods did not rely on formalized provability, and had only a superficial resemblance to Gödel's work . Gödel read the paper but found it deeply flawed, and his response to Finsler laid out concerns about the lack of formalization (). Finsler continued to argue for his philosophy of mathematics, which eschewed formalization, for the remainder of his career.


title: Gödel's incompleteness theorems

text: In September 1931, Ernst Zermelo wrote to Gödel to announce what he described as an "essential gap" in Gödel's argument (). In October, Gödel replied with a 10-page letter (, ), where he pointed out that Zermelo mistakenly assumed that the notion of truth in a system is definable in that system (which is not true in general by Tarski's undefinability theorem). But Zermelo did not relent and published his criticisms in print with "a rather scathing paragraph on his young competitor" (). Gödel decided that to pursue the matter further was pointless, and Carnap agreed (). Much of Zermelo's subsequent work was related to logics stronger than first-order logic, with which he hoped to show both the consistency and categoricity of mathematical theories.


title: Gödel's incompleteness theorems

text: Ludwig Wittgenstein wrote several passages about the incompleteness theorems that were published posthumously in his 1953 "Remarks on the Foundations of Mathematics", in particular one section sometimes called the "notorious paragraph" where he seems to confuse the notions of "true" and "provable" in Russell's system. Gödel was a member of the Vienna Circle during the period in which Wittgenstein's early ideal language philosophy and Tractatus Logico-Philosophicus dominated the circle's thinking. There has been some controversy about whether Wittgenstein misunderstood the incompleteness theorem or just expressed himself unclearly. Writings in Gödel's Nachlass express the belief that Wittgenstein misread his ideas.


title: Gödel's incompleteness theorems

text: Multiple commentators have read Wittgenstein as misunderstanding Gödel , although Juliet Floyd and , as well as have provided textual readings arguing that most commentary misunderstands Wittgenstein. On their release, Bernays, Dummett, and Kreisel wrote separate reviews on Wittgenstein's remarks, all of which were extremely negative . The unanimity of this criticism caused Wittgenstein's remarks on the incompleteness theorems to have little impact on the logic community. In 1972, Gödel stated: "Has Wittgenstein lost his mind? Does he mean it seriously? He intentionally utters trivially nonsensical statements" , and wrote to Karl Menger that Wittgenstein's comments demonstrate a misunderstanding of the incompleteness theorems writing:


title: Gödel's incompleteness theorems

text: Since the publication of Wittgenstein's "Nachlass" in 2000, a series of papers in philosophy have sought to evaluate whether the original criticism of Wittgenstein's remarks was justified. argue that Wittgenstein had a more complete understanding of the incompleteness theorem than was previously assumed. They are particularly concerned with the interpretation of a Gödel sentence for an ω-inconsistent system as actually saying "I am not provable", since the system has no models in which the provability predicate corresponds to actual provability. argues that their interpretation of Wittgenstein is not historically justified, while argues against Floyd and Putnam's philosophical analysis of the provability predicate. explores the relationship between Wittgenstein's writing and theories of paraconsistent logic.


title: Gödel's incompleteness theorems

text: None of the following agree in all translated words and in typography. The typography is a serious matter, because Gödel expressly wished to emphasize "those metamathematical notions that had been defined in their usual sense before . . ." . Three translations exist. Of the first John Dawson states that: "The Meltzer translation was seriously deficient and received a devastating review in the "Journal of Symbolic Logic"; "Gödel also complained about Braithwaite's commentary . "Fortunately, the Meltzer translation was soon supplanted by a better one prepared by Elliott Mendelson for Martin Davis's anthology "The Undecidable" . . . he found the translation "not quite so good" as he had expected . . . [but because of time constraints he] agreed to its publication" (ibid). (In a footnote Dawson states that "he would regret his compliance, for the published volume was marred throughout by sloppy typography and numerous misprints" (ibid)). Dawson states that "The translation that Gödel favored was that by Jean van Heijenoort" (ibid). For the serious student another version exists as a set of lecture notes recorded by Stephen Kleene and J. B. Rosser "during lectures given by Gödel at to the Institute for Advanced Study during the spring of 1934" (cf commentary by and beginning on p. 41); this version is titled "On Undecidable Propositions of Formal Mathematical Systems". In their order of publication:


title: Rocky (franchise)

text: Rocky is an American sports drama multimedia franchise created by Sylvester Stallone, which began with the eponymous 1976 film and has since become a cultural phenomenon, centered on the boxing career of Rocky Balboa. The franchise has been expanded into various films and, with a total worldwide gross of $1.7 billion, is one of the highest-grossing media franchises.


title: Rocky (franchise)

text: The original film ("Rocky") was written by Stallone and directed by John G. Avildsen, and was followed by the sequels "Rocky II" (1979), "Rocky III" (1982), "Rocky IV" (1985), "Rocky V" (1990), and "Rocky Balboa" (2006), all of which were written and directed by Stallone, except for "Rocky V" (1990), for which director John G. Avildsen was brought back, and distributed by Metro-Goldwyn-Mayer. After the sixth film, which was envisioned as the conclusion of the franchise, the spin-off and sequel films "Creed" (2015) and "Creed II" (2018), starring Michael B. Jordan as boxer Adonis Creed, were released; they were directed by Ryan Coogler and Steven Caple Jr. and also starred Stallone.


title: Rocky (franchise)

text: The franchise will continue with a third "Creed" film is scheduled for release in 2023, while a seventh "Rocky" movie, and a prequel television series are in development.


title: Rocky (franchise)

text: The first film, "Rocky III", and "Creed" were nominated for Academy Awards, with the first winning Best Picture, Best Director for Avildsen, and Best Film Editing, and is considered one of the greatest sports films of all time. Stallone was nominated for Best Actor and Best Supporting Actor for his performance in the first film and "Creed", respectively. "Rocky" has influenced landmarks and popular culture; the entrance to the Philadelphia Museum of Art have become known as the Rocky Steps while phrases like "Yo Adrian!" and "If he dies, he dies" have become part of lexicon or widely memed. Balboa is also considered one of the most iconic fictional characters, and the franchise is closely linked with its successful theme songs, including "Gonna Fly Now", "Eye of the Tiger", and "Burning Heart".


title: Rocky (franchise)

text: Rocky Balboa (Sylvester Stallone) is a small-time boxer who seems to be going nowhere in life, as he works day-in and day-out as a collector for a loan shark and fights in sleazy clubs for low pay rewards, for which Rocky is mocked and told that he's nothing but a 'bum', especially by gym trainer Mickey Goldmill (Burgess Meredith). At the same time, Rocky successfully courts Adrian Pennino (Talia Shire), a painfully shy woman with an alcoholic brother, Paulie (Burt Young). But when the world heavyweight champion boxer Apollo Creed (Carl Weathers) chooses Rocky at random as his opponent in a title fight, Rocky realizes he now has the chance to prove that he's not worthless. With Adrian's support, sponsorship offers with the help from Paulie, and Mickey becoming his trainer and manager, Rocky is determined to go the distance with Creed and fights for his self-respect.


title: Rocky (franchise)

text: Soon after proving himself, even with a split decision loss to Apollo Creed, Rocky expects the good life to follow. He marries Adrian and begins spending the money he earned from the match. But after he fails at both endorsements and a series of low wage jobs, Rocky realizes the only way he can survive is to begin boxing again. Creed on the other hand, faces criticism from fans to overcome the fight. As a result, he publicly taunts Rocky into a rematch, for which Rocky trains once again with Mickey. In the fifteenth round, Rocky knocks Creed to the ground, falling to the canvas himself in the process. Both fighters struggle to get to their feet, but only Rocky is successful. For the first time, Rocky is declared the World Heavyweight Champion; even Apollo shows respect for him as he regains his support from fans for doing a fair fight.


title: Rocky (franchise)

text: After winning the heavyweight title, Rocky takes advantage of his newfound wealth and fame, appearing in multiple advertisements and television programs, and relishing his new celebrity status. After defending the title multiple times, he is prepared to retire, but the number one contender, James "Clubber" Lang (played by Mr. T), challenges Rocky publicly. Rocky, after dealing with Mickey's heart attack before the fight, is overpowered by the stronger, hungrier Lang and is knocked out in the second round. Mickey dies after the fight, and former rival Apollo Creed steps in, training Rocky to fight in Creed's old style (in his old Los Angeles gym) and to use more guile and skill. In the rematch, Rocky outboxes Lang, tiring the stronger fighter out and eventually knocking him out in the third round. After the fight, Apollo calls in his "favor" for training Rocky, which is a one-on-one match between the two of them with no cameras, no media, just man vs. man in the gym. The film ends as they each throw their first punch.


title: Rocky (franchise)

text: After winning back his title from Clubber Lang, Rocky continues to live the good life, now that he and Apollo are good friends. However, a new fighter from the USSR, Ivan Drago (played by Dolph Lundgren), has emerged and challenges Rocky to an exhibition match. Apollo fights instead, and the beating he takes from Drago ends with Apollo dying in Rocky's arms in the ring as Drago coldly watches. To avenge Apollo, Rocky challenges Drago to a match, which is to be held on Christmas Day in Moscow. In a montage replete with symbolism, Rocky is shown training in a remote cabin in Siberia with the help of Creed's old trainer Duke, his brother-in-law Paulie, and eventually Adrian, doing exercises such as chopping wood, lifting rocks, running in the snow and climbing a mountain, while Drago is seen in an advanced training facility running on treadmills, utilizing weightlifting machines and injecting steroids to boost his strength. During the fight, Rocky takes the worst beating of his life but refuses to fall. He eventually wins over the foreign crowd with his display of courage and determination, and he knocks Drago out with seconds left in the final round.


title: Rocky (franchise)

text: In the aftermath of his fight with Ivan Drago, Rocky Balboa is diagnosed with brain damage and is forced to retire from the ring. Additionally, the Balboa fortune is all gone due to an unscrupulous accountant. Rocky's family returns to their old neighborhood and Adrian returns to the pet store she used to work at, while (in a subplot) Rocky (Robert) Jr. (played by Sylvester Stallone's real son Sage) deals with bullying at his school and Rocky re-opens Mickey's old gym. While training other boxers, Rocky meets a young, hungry boxer named Tommy Gunn (played by real-life fighter Tommy Morrison) and begins training him which results in a strained relationship with Robert. Unfortunately, as Tommy begins his rise to fame under Rocky's wing, a sleazy fight promoter named George Washington Duke convinces Tommy that Rocky is holding him back and Tommy leaves Rocky for Duke. After Tommy wins the heavyweight title, he makes a short speech thanking Duke, and is met with jeers and the familiar chant of "Rocky" from the crowd. Seething from this insult, as well as being called by publicists as "Rocky's robot" and being told that he's "no Rocky Balboa", Tommy decides to seek out his former mentor for a final showdown. Rocky starts to walk away from the public challenge, but Paulie decides to let Tommy have a piece of his mind about how Tommy has treated Rocky, and Tommy proceeds to punch Paulie. Rocky then challenges Tommy outside and the two proceed in a violent bare-knuckle street brawl, which Rocky wins. Rocky then proceeds to punch Duke for harassing him. In the end, Rocky and Robert reconcile as they run up the steps as father and son.


title: Rocky (franchise)

text: In "Rocky Balboa", twenty years have passed since his final fight with his former protégé, Tommy "The Machine" Gunn. Long retired, Rocky Balboa still staggers around an ever-changing world; his son is grown and distant, Paulie is working back at the meat plant, and Rocky's wife Adrian has died. Rocky has opened a restaurant named after his wife, which he stocks with mementos of his prime as he tells his old fight stories to customers. But when a computer-simulated fight on ESPN depicting a bout between a young Rocky Balboa and the current champion, Mason Dixon (Antonio Tarver) reignites interest in the faded boxer, Rocky discovers he has not lost his fighting spirit and considers an opportunity to prove himself in the ring again. Rocky almost wins the fight but loses in a split decision just like the first film. Rocky is last seen visiting his wife's grave saying, "Yo Adrian, we did it".


title: Rocky (franchise)

text: Adonis "Donnie" Johnson (Michael B. Jordan), the illegitimate son of the late former heavyweight champion Apollo Creed, tracks down Rocky Balboa at Adrian's and asks Rocky to become his trainer. Rocky is reluctant, but eventually agrees. When word gets out that Donnie is Creed's illegitimate son, the handlers of world light heavyweight champion "Pretty" Ricky Conlan, who is forced into retirement by an impending prison term, offer to make Donnie the latter's final challenger—provided that he change his name to Adonis Creed. Donnie balks at first, wanting to forge his own legacy, but he eventually agrees. While helping Donnie train, Rocky learns he has non-Hodgkin's lymphoma. He is initially unwilling to undergo chemotherapy because it wasn't enough to save Adrian from ovarian cancer. Donnie persuades Rocky to seek treatment and continues training while Rocky recovers. In a battle reminiscent of Apollo and Rocky's first fight, Donnie fights Conlan in Conlan's hometown of Liverpool and surprises almost everyone by going the distance and pushing Conlan to his limit. Conlan wins by split decision, but Donnie wins the respect of Conlan and the crowd, with Conlan calling Donnie the future of the light heavyweight division. Returning to Philadelphia, Donnie and a recovering Rocky both make their way up the Philadelphia Museum steps and look at the skyline.


title: Rocky (franchise)

text: Three years after losing the fight to "Pretty" Ricky Conlan, Adonis Johnson Creed wins the World Heavyweight Championship and proposes to his girlfriend, Bianca Taylor (Tessa Thompson). Meanwhile, three decades since the death of Apollo Creed and his loss to Rocky Balboa, Ivan Drago is training his son, Viktor Drago (Florian Munteanu), to reclaim their honor by having Viktor publicly challenge Adonis for the World Heavyweight Title. Rocky is reluctant to train Adonis, fearing Adonis will meet the same fate as his father, but Adonis accepts the challenge without him and is badly injured during the fight. Because of Viktor's disqualification, Adonis retains the title. Viktor demands a rematch, while he is being promoted by his father's supporters who originally abandoned Drago, including Drago's ex-wife, Ludmilla (Brigitte Nielsen). Viktor is subjected to vicious training sessions in preparation for the rematch. Rocky comes to Adonis' aid and decides to train him in the Southern California desert, with a different tactic for Adonis to take in Viktor's powerful punches. The rematch is set in Moscow, and Adonis is able to take Viktor's powerful punches, while leaving Viktor exhausted. Viktor's supporters and his mother leave during the fight with doubt that Viktor will win. Drago seeing his son taking Adonis’ punches without fighting back, throws in the towel and assures his son that he is more important to him and that he will not abandon him like he was. While Rocky later reunites with his son, Robert (Milo Ventimiglia) and meets his grandson, Adonis goes to his late father's grave, making peace with his late father and honoring his legacy.


title: Rocky (franchise)

text: In December 2018, Sylvester Stallone confirmed that there are ongoing discussions regarding a third "Creed" film. That same month, professional boxer Deontay Wilder announced plans to begin a career in acting, stating specifically that he wants to play James "Clubber" Lang's son in "Creed III". Stallone and Michael B. Jordan expressed mutual interest in such a character featuring in the plot of the next installment, while Stallone stated that he wasn't opposed to the casting choice. "Creed III" was later officially confirmed to be in development. Producers expressed interest in having Jordan serve as director, with Irwin Winkler stating that he had personally offered the position to the actor.


title: Rocky (franchise)

text: In February 2020, Zach Baylin signed on as screenwriter with Jordan confirmed to reprise his role as Adonis "Donnie" Creed. By October of the same year, Jordan entered early negotiations to be the film's director in addition to his starring role. By March 2021 Jordan officially signed on as director of the project, marking the directorial debut for the star as a filmmaker, with a script co-written by Zach Baylin and Keenan Coogler, from an original story written by Ryan Coogler. Irwin Winkler, Charles Winkler, William Chartoff, David Winkler, Jonathan Glickman, Jordan and Ryan Coogler serve as co-producers. The project is a joint-venture production between MGM, Chartoff-Winkler Productions, Proximity Media, and United Artists Releasing. "Creed III" is scheduled to be released on March 3, 2023. Stallone later announced that he will not appear as Balboa in the film.


title: Rocky (franchise)

text: In May 2019 at the Cannes Film Festival, Sylvester Stallone said that he had another story about Rocky Balboa. By July, Stallone confirmed that a sequel/follow-up to the current film series is in development. The project will be a joint-production venture between Winkler Films Production and MGM. Stallone will serve as writer in addition to starring in the film.


title: Rocky (franchise)

text: Conceptualized as an epilogue story, the film is said to be about Rocky befriending a young fighter who is a foreigner, stuck illegally in the United States. Stallone states: "Rocky meets a young, angry person who got stuck in this country when he comes to see his sister. He takes him into his life, and unbelievable adventures begin, and they wind up south of the border. It's very, very timely". By May 2020, Stallone said that he is still working on the film, though it has not yet been officially green-lit by the studio. However in a November 2021 Instagram post, Stallone expressed doubt about the film being greenlit, due to his increasingly sour relationship with Irwin Winkler. In November 2022, Stallone confirmed that the studio wants another "Rocky" film, but that negotiations to attain part of the rights to the character from the producers stalled development. He further stated that he is writing the script and that if the studio likes his work, the film will be made.


title: Rocky (franchise)

text: In November 2021, Dolph Lundgren revealed that there are developments for a film centered around his character Ivan Drago. By July 2022, the project was officially announced by MGM with Robert Lawton serving as screenwriter on the project. Lawton had previously completed a spec script for a movie pitch centered around the making of the first "Rocky" film. Though the studio did not choose to develop that script, studio executives were impressed enough to hire the writer for the "Drago" film. The screenplay is stated to be centered around Ivan Drago's backstory. The announcement of the spin-off movie, was met with criticism by franchise creator, Stallone. He stated that while he has tried to attain part of the rights to a franchise he was instrumental in creating, the producers have been "exploiting" his characters while trying to keep him out of the creative team. Following Stallone's response, Lundgren stated that he immediately reached out to his friend, explaining that when the project had been presented to him he had been informed that Stallone would be involved as a producer, as well as appearing in the movie; the actor further stated that the project isn't officially green-lit yet. While Stallone's reactions were perceived as uncharacteristic to his public image, he later stated that he has since discussed the potential movie with Lundgren.


title: Rocky (franchise)

text: In July 2019, Sylvester Stallone announced that there are "ongoing discussions" about a "Rocky" prequel television series, with the project intended to premiere on a streaming service platform. Film series producer, Irwin Winkler however, is said to be hesitant in making a television series with concerns regarding how the story would translate to television. As an official production start has not yet been announced, Stallone stated: "There was some conflict there, yes. ...so there was a big bone of contention".


title: Rocky (franchise)

text: In March 2021, Stallone revealed he had started working on the script for a prequel series that will take place during the 1960s, with intention for the show to be developed as a streaming exclusive with multiple seasons, each consisting of ten episodes. By May, Stallone said that though there are "certain complications" with developing the series, but that he would "keep punching". Despite this setback, later that month as the acquisition of MGM by Amazon for $8.45 billion had initiated, Mike Hopkins (Senior Vice President of Amazon Prime Video and Amazon Studios) identified the "Rocky" franchise as one of the major assets in the purchase. He also stated that the "value behind this deal is the treasure trove of IP in the deep catalog that we plan to reimagine and develop together with MGM's talented team".


title: Rocky (franchise)

text: In November 2022, Stallone confirmed that his prequel script which was previously turned down by producers is once again in development for a streaming company. He stated that Amazon's new inclusion in the franchise, is responsible for the progress of the project.


title: Rocky (franchise)

text: At the 49th Academy Awards, "Rocky" was nominated for ten Academy Awards. Sylvester Stallone was nominated for the Academy Award for Best Actor, and Best Original Screenplay, with Talia Shire also being nominated for Best Actress, and both Burgess Meredith and Burt Young being nominated for Best Supporting Actor. "Gonna Fly Now" was nominated for Best Original Song, and "Rocky" itself was nominated for Best Sound Editing, and won Best Picture, Best Director for John G. Avildson, and Best Film Editing. At the 55th Academy Awards, the song "Eye of the Tiger" from "Rocky III" was nominated for Best Original Song.


title: Rocky (franchise)

text: On December 7, 2010, Stallone was inducted into the International Boxing Hall of Fame and Museum, for paying tribute to boxers in writing and creating the underdog character of Rocky. In January 2016, Stallone won the Golden Globe Award for Best Supporting Actor for his role in "Creed". At the 88th Academy Awards, Stallone was nominated for the Academy Award for Best Supporting Actor for his performance in "Creed", the film's only nomination. In total, the "Rocky" series has received twelve Academy Awards nominations, winning three.


title: Rocky (franchise)

text: A Broadway musical was written by Stephen Flaherty and Lynn Ahrens (lyrics and music), with the book by Thomas Meehan, based on the film. The musical premiered in Hamburg, Germany in October 2012. Performances commenced at the Winter Garden Theater on Broadway on February 11, 2014, and officially opened on March 13.


title: Rocky (franchise)

text: "Rocky" is featured in the 2017 documentary "John G. Avildsen: King of the Underdogs" about Academy Award-winning "Rocky" director John G. Avildsen, directed and produced by Derek Wayne Johnson.


title: Rocky (franchise)

text: Stallone later hand-picked Johnson to direct and produce a documentary on the making of the original "Rocky", entitled "40 Years of Rocky: The Birth of a Classic", which was released in 2020. The documentary features Stallone narrating behind-the-scenes footage from the making of the film.


title: Rocky (franchise)

text: A director's cut edition of "Rocky IV" was originally scheduled to commemorate its 35th anniversary. Ongoing editing ultimately pushed the release date, with Stallone finishing his editing sometime in January 2021. In all, approximately 38 minutes of previously unreleased footage were added to the film, including significant extensions of both fight scenes and the Apollo Creed funeral scene. One reported cut was that of Paulie's robot. Robert Doornick, founder of International Robotics and the voice of the robot, commented that Stallone cut all of the robot scenes in the director's cut to save money on royalty fees that were given to Doornick in the original cut.


title: Rocky (franchise)

text: The cut missed its original November 27, 2020 release, though by February 2021 Stallone stated that they were in the final stages of completing the project. His cut of the film was finished in April. This version had a one-night limited theatrical release on November 11, 2021 and was additionally released in digital formats the following day.


title: Rocky (franchise)

text: The 2016 film "Chuck" depicts Chuck Wepner, his 1975 title fight with the heavyweight champion, Muhammad Ali, and the fight's influence on the screenplay for "Rocky".


title: Rocky (franchise)

text: In the January 2022 issue of "Empire", published in November 2021, Ralph Macchio revealed that he had previously been approached about a potential "Rocky" and "The Karate Kid" crossover film in 2012, to be directed by John G. Avildsen which would follow Daniel LaRusso's daughter and Milo Ventimiglia's Rocky Balboa Jr. "get[ting] together and open[ing] a "dojo"". Describing the concept as "awful", Macchio declined reprising his role and the project subsequently entered development hell, before being abandoned in favor of "Creed" and "Cobra Kai" individually.


title: Polyester

text: Polyester is a category of polymers that contain the ester functional group in every repeat unit of their main chain. As a specific material, it most commonly refers to a type called polyethylene terephthalate (PET). Polyesters include naturally occurring chemicals, such as in plants and insects, as well as synthetics such as polybutyrate. Natural polyesters and a few synthetic ones are biodegradable, but most synthetic polyesters are not. Synthetic polyesters are used extensively in clothing.


title: Polyester

text: Polyester fibers are sometimes spun together with natural fibers to produce a cloth with blended properties. Cotton-polyester blends can be strong, wrinkle- and tear-resistant, and reduce shrinking. Synthetic fibers using polyester have high water, wind and environmental resistance compared to plant-derived fibers. They are less fire-resistant and can melt when ignited.