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Gleichschaltung

The Nazi term Gleichschaltung or "coordination" was the process of Nazification by which Adolf Hitler — leader of the Nazi Party in Germany — successively established a system of totalitarian control and coordination over all aspects of German society "from the economy and trade associations to the media, culture and education". Although the Weimar Constitution remained nominally in effect until Germany's surrender following World War II, near total Nazification had been secured by the 1935 resolutions approved during the Nuremberg Rally, when the symbols of the Nazi Party and the state were fused (see Flag of Nazi Germany) and German Jews were deprived of their citizenship (see Nuremberg Laws). The tenets of Gleichschaltung also applied to territories occupied by the Nazis.

Terminology
Gleichschaltung is a compound word that comes from the German words gleich (same) and schaltung (circuit) and was derived from an electrical engineering term meaning that all switches are put on the same circuit so that all can be activated by throwing a single master switch. Its first use is credited to Reich Justice Minister Franz Gürtner. It has been variously translated as "coordination", "Nazification of state and society", "synchronization", and "bringing into line". English texts often use the untranslated German word to convey its unique historical meaning. In their seminal work on National Socialist vernacular, Nazi-Deutsch/Nazi-German: An English Lexicon of the Language of the Third Reich, historians Robert Michael and Karin Doerr define Gleichschaltung as: "Consolidation. All of the German Volk's social, political, and cultural organizations to be controlled and run according to Nazi ideology and policy. All opposition to be eliminated." This accords with the general description provided by historian Jane Caplan, who characterized the term as "the coordination of German institutions into a cohesive, Nazified whole."

Legal basis
The Nazis were able to put Gleichschaltung into effect due to multiple legal measures enacted by the Reich government during the 20 months following 30 January 1933, when Adolf Hitler became Chancellor of Germany.

 * Reichstag Fire Decree. The day after the Reichstag fire, President of Germany Paul von Hindenburg, acting at Hitler's request and based on the emergency powers in article 48 of the Weimar Constitution, issued the Reichstag Fire Decree (28 February 1933). This decree, formally titled "Decree of the Reich President for the Protection of People and State," suspended most civil rights and civil liberties enshrined in the constitution, including the right of habeas corpus, freedom of speech, press, assembly and privacy of communications. This allowed for the arrest of political adversaries, mostly Communists, and for terrorizing of other voters by the Sturmabteilung (SA) (Nazi paramilitary branch) before the upcoming election. In this atmosphere, the general election of the Reichstag took place on 5 March 1933. The Nazis had hoped to win an outright majority and push aside their coalition partners, the German National People's Party (DNVP). However, the Nazis won only 43.9 percent of the vote, short of a majority and well below the number needed to amend the federal constitution.

 * Enabling Act. When the newly elected Reichstag convened – not including the Communist delegates whose participation in politics had been banned – it passed the Enabling Act (23 March 1933). This law, formally titled "Law to Remedy the Distress of the People and the Reich," gave the government (the Reich Chancellor and his cabinet) the right to enact laws for a period of four years without the involvement of the Reichstag or the Reich President. Under certain circumstances, these laws could "deviate from the Constitution." As a constitutional amendment, it required a two-thirds majority for passage. Even with the proscription of the Communists, the Nazis and their ally the DNVP were still short of this number. Through intimidation of deputies by surrounding the Reichstag with a cordon of SA members, and through promises of religious freedom protections to the Catholic Centre Party, the required supermajority was obtained. With only the SPD in opposition, the Enabling Act passed 444–94. For all intents and purposes, the entire Weimar Constitution was rendered void. It formed the cornerstone on which Hitler circumvented the constitutional framework of the Weimar Republic and was able to impose his will on the nation by decree. For all intents and purposes, it converted his government into a legal dictatorship.

 * Provisional Law on the Coordination of the States with the Reich. Enacted by the Reich government using the Enabling Act, the "Provisional Law on the Coordination of the States with the Reich" (31 March 1933) dissolved the sitting parliaments of all German states except the recently elected Prussian parliament, which the Nazis already controlled. It also ordered the state parliaments reconstituted based on the state votes cast in the 5 March Reichstag election (except for Communist seats, which were not filled). Under this provision, the Nazis and their DNVP partners were able to attain working majorities in all the parliaments. It further mandated the simultaneous dissolution of all state parliaments whenever the Reichstag was dissolved. It also gave the state governments the same powers to enact legislation that the Reich government possessed under the Enabling Act.

 * Second Law on the Coordination of the States with the Reich. In order to further extend their power over the German states, the Reich government enacted the "Second Law on the Coordination of the States with the Reich" (7 April 1933). This measure deployed one Reichsstatthalter (Reich Governor) in each state. These officers, appointed by the President on the recommendation of the Chancellor, were responsible to Interior Minister Wilhelm Frick and were intended to act as local proconsuls in each state, with near-complete control over the state governments. They were empowered to preside over meetings of the state government, appoint and dismiss the state minister-president as well as other high officials and judges, dissolve the state parliament, call new elections and promulgate state laws. The law conferred the office of Reichsstatthalter in Prussia on the Reich Chancellor himself.

 * Law for the Restoration of a Professional Civil Service. Another measure of Nazi Gleichschaltung was the enactment of the "Law for the Restoration of a Professional Civil Service" (7 April 1933), which mandated the "co-ordination" of the civil service – which in Germany included not only bureaucrats, but also schoolteachers and professors, judges, prosecutors, and other professionals – at the federal, state and municipal level, and authorized the removal of Jews and Communists from these positions, with limited exceptions for those who had fought in the First World War or had lost a father or son in combat.

 * Law on the Trustees of Labour. On 2 May 1933, trade union offices were attacked and occupied by SA stormtroopers. The offices were closed, their newspapers shuttered, funds confiscated and leaders arrested. In order to impose coordination on private sector workers, the Law on the Trustees of Labour (19 May 1933) established the position of Trustee of Labour, each of which was assigned to one of thirteen large economic areas (Wirtschaftsgebiete). They were charged with ensuring industrial peace and regulating employment contracts, including the setting of wages and the resolution of employer–employee disputes. This effectively supplanted collective bargaining and strikes, as the trustees were authorized to impose legally-binding settlements.

 * Law Against the Formation of Parties. The Communist Party had effectively been outlawed in all but name by the Reichstag Fire Decree, and was completely banned from 6 March. Following additional months of violence and intimidation against the Social Democratic Party, the government seized all its assets, and banned it outright on 22 June 1933, canceling all SPD electoral mandates in both the Reichstag and the state parliaments. By early July, all other parties, even the Nazis' erstwhile allies the DNVP, had been intimidated into dissolving themselves rather than face arrests and concentration camp imprisonment. Thus the DNVP (27 June) the German State Party (28 June), the Bavarian People's Party (4 July), the German People's Party (4 July) and the Centre Party (5 July) all formally disbanded. The "Law Against the Formation of Parties" (14 July 1933) declared the NSDAP as the country's only legal political party, formalizing what had already been accomplished through the campaign of Nazi terror and the complete capitulation of the opposition.

 * Law to Secure the Unity of Party and State. With the Nazi Party as the only remaining legal party, Hitler now sought to extend the Party's grasp over all the levers of state power and administration through the "Law to Secure the Unity of Party and State" (1 December 1933), which was enacted by the Reich government under the provisions of the Enabling Act. The Nazi Party was legally established as a public corporation with jurisdiction over its members. The Party and the SA became official organs of the German Reich. The Deputy Führer of the Party (Rudolf Hess) and the Stabschef of the SA (Ernst Röhm), were made ex officio members of the Reich government as ministers without portfolio, further interlocking the leadership of the Party and State. Government agencies, including public safety and law enforcement authorities such as the police, public prosecutors and the courts, were obligated to provide the Party and the SA with administrative and legal information and investigatory assistance. Party courts (Parteigerichte) received the status of official legal institutions of the State and any crime committed against the Party now was considered a crime against the State. These courts were now authorized to impose detention or imprisonment as punishment.

 * Law on the Reconstruction of the Reich. All the state parliaments had been dissolved (along with the Reichstag) on 14 October 1933, and while new Reichstag elections took place on 12 November, no new state parliamentary elections were scheduled. Now, on the one-year anniversary of coming to power, the Reich government passed through the Reichstag, by a unanimous vote, the "Law on the Reconstruction of the Reich" (30 January 1934). This was one of only seven laws passed by the Reichstag in the 19 sessions held during the entire Nazi regime, as opposed to 986 laws enacted solely by the Reich government under the authority of the Enabling Act. The law, in the form of a constitutional amendment, formally did away with the concept of a federal republic. The state parliaments were abolished altogether and state sovereignty passed to the Reich government. The states, though not themselves eliminated, were reduced to mere administrative bodies subordinated to the Reich, effectively converting Germany into a highly centralized unitary state. By destroying the autonomy of the historic German states, Hitler achieved what Bismarck, Wilhelm II and the Weimar Republic had never dared to attempt.

 * Law on the Abolition of the Reichsrat. Within two weeks of the abolition of the state parliaments, the Reich government enacted the "Law on the Abolition of the Reichsrat" (14 February 1934) formally abolishing the Reichsrat, the second chamber of the national parliament that represented the states. This was a clear violation of the Enabling Act. While Article 2 of the Enabling Act allowed the government to pass laws that deviated from the Constitution, it explicitly protected the existence of the Reichstag and Reichsrat.

 * Law Concerning the Head of State of the German Reich. With Reich President von Hindenburg fatally ill, the Reich government enacted the "Law Concerning the Head of State of the German Reich" (1 August 1934). This law was signed by the entire Reich cabinet. It combined the office of Reich President with that of Reich Chancellor under the title of "Führer and Reich Chancellor," and was drawn up to become effective on the death of the Reich President, which occurred the next day. Again, this flagrantly violated Article 2 of the Enabling Act, which forbade any actions interfering with the office of the Reich President. With this law, Hitler became not only Germany's head of state, but also the commander-in-chief of the armed forces. It also removed the last remedy by which Hitler could be legally removed from office, and with it all checks on his power.

Coordination of the German Länder

When Hitler was appointed Reich Chancellor on 30 January 1933, the Nazi Party had control of only five of the seventeen German Länder (states). However, the Nazis acted swiftly to eliminate any potential centers of opposition arising in the remaining states. Immediately following the Reichstag election of 5 March 1933, the central government began in earnest its campaign to take over the state governments that it did not yet control, and within a very short period they achieved dominance over the administration in every state.

"The pattern was in each case similar: pressure on the non-Nazi state governments to place a National Socialist in charge of the police; threatening demonstrations from SA and SS troops in the big cities; the symbolic raising of the swastika banner on town halls; the capitulation with hardly any resistance of the elected governments; the imposition of a Reich Commissar under the pretext of restoring order … Despite the semblance of legality, the usurpation of the powers of the Länder by the Reich was a plain breach of the Constitution. Force and pressure by the Nazi organizations themselves – political blackmail – had been solely responsible for creating the 'unrest' that had prompted the alleged restorations of 'order'. The terms of the emergency decree of 28 February provided no justification since there was plainly no need for defence from any 'communist acts of violence endangering the state'. The only such acts were those of the Nazis themselves."

The following table presents an overview of the process of Gleichschaltung as it was applied to the Nazification of the German Länder governments. While, strictly speaking, the Gleichschaltung process did not start until after the Nazi seizure of power at the Reich level at the end of January 1933, the table also presents earlier Nazi Party successes in infiltrating and taking charge of several German state administrations during 1930–1932. In most of these instances, they took the portfolio of the state interior ministries from which they controlled the police, installing Nazi adherents and purging opponents.

Most coalition cabinets that the Nazis formed were with the participation of their conservative nationalist ally, the German National People's Party (DNVP). The "Law Against the Founding of New Parties" (14 July 1933) banned all parties except the Nazi Party. The DNVP members of the remaining coalition cabinets eventually either joined the Party or were replaced by Nazis, resulting in one-party government in all the Länder.

Propaganda and societal integration
One of the most critical steps towards Gleichschaltung of German society was the introduction of the "Ministry of Public Enlightenment and Propaganda" under Joseph Goebbels in March 1933 and the subsequent steps taken by the Propaganda Ministry to assume complete control of the press and all means of social communication. This included oversight of newspapers, magazines, films, books, public meetings and ceremonies, foreign press relations, theater, art and music, radio, and television. To this end, Goebbels said:

"[T]he secret of propaganda [is to] permeate the person it aims to grasp, without his even noticing that he is being permeated. Of course propaganda has a purpose, but the purpose must be concealed with such cleverness and virtuosity that the person on whom this purpose is to be carried out doesn't notice it at all."

This was also the purpose of "co-ordination": to ensure that every aspect of the lives of German citizens was permeated with the ideas and prejudices of the Nazis. From March to July 1933 and continuing afterward, the Nazi Party systematically eliminated or co-opted non-Nazi organizations that could potentially influence people. Those critical of Hitler and the Nazis were suppressed, intimidated, or murdered.

"Every national voluntary association, and every local club, was brought under Nazi control, from industrial and agricultural pressure groups to sports associations, football clubs, male voice choirs, women's organizations—in short, the whole fabric of associational life was Nazified. Rival, politically oriented clubs or societies were merged into a single Nazi body. Existing leaders of voluntary associations were either unceremoniously ousted, or knuckled under of their own accord. Many organizations expelled leftish or liberal members and declared their allegiance to the new state and its institutions. The whole process ... went on all over Germany. ... By the end, virtually the only non-Nazi associations left were the army and the Churches with their lay organizations."

For example, in 1934, the government founded the Deutscher Reichsbund für Leibesübungen, later the Nationalsozialistischer Reichsbund für Leibesübungen, as the official sports governing body. All other German sport associations gradually lost their freedom and were coopted into it. Besides sports, another more important part of the "co-ordination" effort was the purging of the civil service, both at the Federal and state level. Top Federal civil servants—the State Secretaries—were largely replaced if they were not sympathetic to the Nazi program, as were the equivalent bureaucrats in the states, but Nazification took place at every level. Civil servants rushed to join the Nazi Party, fearing they would lose their jobs if they did not. At the local level, mayors and councils were terrorized by Nazi stormtroopers of the SA and SS into resigning or following orders to replace officials and workers at local public institutions who were Jewish or belonged to other political parties.

The Gleichschaltung also included the formation of various organizations with compulsory membership for segments of the population, particularly the youth of Germany. Boys first served as apprentices in the Pimpfen (cubs), beginning at the age of six, and at age ten, entered the Deutsches Jungvolk (Young German Boys) and served there until joining the Hitler Youth proper at age fourteen. Boys remained there until age eighteen, at which time they entered into the Arbeitsdienst (Labor Service) and the armed forces. Girls became part of the Jungmädel (Young Maidens) at age ten and at age fourteen were enrolled in the Bund Deutscher Mädel (League of German Maidens). At eighteen, BDM members generally went to the eastern territory for their Pflichtdienst, or Landjahr, a year of labor on a farm. By 1940, membership in the Hitler Youth numbered some eight million.

Coordination of the trade union movement
The German trade union movement had a long history, dating to the mid-nineteenth-century. At the time of the Weimar Republic, its largest grouping was the General German Trade Union Federation (ADGB). This was an umbrella organization that was formed in July 1919 and was originally composed of 52 unions with about 8 million workers. It was generally affiliated with the Social Democratic Party (SPD) and was on the left of the political spectrum. In March 1920, it was instrumental in calling a general strike that led to the collapse of the right-wing Kapp Putsch that attempted to overthrow the republic. It was led from January 1921 by Theodor Leipart. Following the economic downturn of 1929, the resulting sharp rise in unemployment caused a large drop-off in membership but, by 1932, it still represented an estimated 3.5 million workers in some 30 unions.

When the Nazis came to power at the end of January 1933, there was some sentiment for a general strike by SPD politicians and trade unionists, but the national leadership was wary of such an action in the face of the worst unemployment crisis the nation had experienced. Though there were some sporadic isolated incidents, no general policy of resistance was undertaken. The Nazis embarked on a policy of violence and intimidation against all their opponents, including the SPD-affiliated trade unions. In an effort to safeguard his organization and its members, Leipart declared the ADBG politically "neutral" within weeks of the Nazis coming to power. Meanwhile, Party leaders convinced conservative elements among the police, the judiciary, prison administrators and civil servants that suppression of the labor movement was justified.

Following the Nazi gains in the Reichstag election of 5 March 1933, violent episodes increased in intensity, with SA stormtroopers ransacking trade union offices, assaulting staff, destroying furniture and equipment, stealing funds and burning documents. By 25 March, union offices in some 45 towns throughout the Reich had been attacked. At this point, the trade unions began to distance themselves from the SPD in an attempt to seek an accommodation with the regime. On 28 April, the ADGB agreed to move toward unification with the conservative Christian and the bourgeois liberal trade union groups, to form a single national labor organization in the new Nazi state. Leipart also supported the announcement by Goebbels that May Day would be celebrated as a public holiday for the first time, a long-sought goal of the labor movement. However, any efforts at reconciliation on the part of the unions were to prove futile, as the Nazis had already begun to plot a complete takeover of the trade union movement, as demonstrated by Goebbels' diary entry of 17 April:

"On 1 May we shall arrange May Day as a grandiose demonstration of the German people's will. On 2 May the trade union offices will be occupied. Coordination in this area too. There might possibly be a row for a few days, but then they will belong to us. We must make no allowances anymore. … Once the trade unions are in our hands the other parties and organizations will not be able to hold out for much longer."

Accordingly, 1 May 1933 was declared the Day of National Labor, a day of parades, speeches and propaganda displays to celebrate the unity of the German labor movement with the nation, featuring SA military bands, swastika flags and fireworks. It culminated with a huge rally and speech by Hitler at Berlin Tempelhof Airport that was attended by over one million people. Many workers, particularly those in state employment, were compelled to participate by threats of dismissal for not attending. The next day, as Goebbels had indicated, the German labor movement was crushed under a wave of unprecedented violence at the hands of SA and SS troops. All SPD-supported trade union offices were occupied, their newspapers and periodicals were discontinued, their banks and credit unions were closed and their assets were confiscated and turned over to the National Socialist Factory Cell Organization. Leipart and other union leaders were taken into "protective custody" and sent to concentration camps. Violence was inflicted on many and, in the most brutal incident, four union officials were beaten to death in Duisburg.

Two days later, other non-SPD-aligned union amalgamations, such as the conservative German National Association of Commercial Employees and the liberal Hirsch-Dunckersche Gewerkvereine, placed themselves under the Action Committee for the Protection of German Labor, headed by Robert Ley, the Stabschef (chief of staff) of the Nazi Party organization. The last remaining union umbrella agency, the Christian-oriented Deutscher Gewerkschaftsbund (1919–1933) was absorbed at the end of June. Throughout this process, there were no strikes, no demonstrations and no protests. Even the Nazis were surprised. The most highly organized and powerful trade union movement in Europe offered no resistance and disappeared virtually overnight.

In its place, the German Labor Front (Deutsche Arbeitsfront or DAF), a Nazi Party organization led by Ley, was established on 10 May. Its goal was coordination of the entire labor force under Nazi leadership. On 19 May, the government enacted the Law on the Trustees of Labour that decreed an end to collective bargaining. It established Trustees of Labour, who were appointed by Hitler and charged with regulating labor contracts and maintaining labor peace. Since their decisions were legally binding, strikes were effectively outlawed.

Strength Through Joy
An all-embracing recreational organization for workers, called Kraft durch Freude ("Strength Through Joy") was set up under the auspices of the German Labor Front. Hobbies were regimented, and all private clubs, whether they be for chess, football, or woodworking, were brought under the control of Strength Through Joy, which also provided vacation trips, skiing, swimming, concerts, and ocean cruises. Some 43 million Germans enjoyed trips via the Strength Through Joy initiative. This effort inspired the idea of Germans acquiring automobiles and the construction of the Autobahn. It was the largest of the many organizations established by the Nazis and a propaganda success. Workers were also brought in line with the party through activities such as the Reichsberufswettkampf, a national vocational competition. Many unemployed people were also drafted into the Reich Labour Service where they were given uniforms and tools and put to work; the disappearance of unemployed people from the streets contributed to the perception that the Nazis were improving the economic conditions of Germany.

Implications
Historian Claudia Koonz explains that the word Gleichschaltung stems from the arena of electricity, where it refers to converting power from alternating current to direct current, which is called "rectification" in English; the word Gleichschaltung translates literally as "phasing". Used in its socio-political sense, Gleichschaltung has no equivalent in any other language. The Nazis also used other similar terms, such as Ausschaltung, which constituted the removal or "switching off" of anyone who stained or soiled the German nation. This seemingly clinical terminology captured both the mechanical and biological meaning for members of German society; as one German citizen visiting London explained, "It means the same stream will flow through the ethnic body politic [Volkskörper]."

Former University of Dresden professor of romance languages, Viktor Klemperer—dismissed from his post for being Jewish in 1935 and who only survived his time in Germany due to being married to a prominent German woman—collected a list of terms employed in everyday speech by the Nazis, which he discussed in his book, LTI – Lingua Tertii Imperii, published in English as The Language of the Third Reich. In this work, Klemperer contends that the Nazis made the German language itself a servant to their ideology through its repetitive use, eventually permeating the very "flesh and blood" of its people. For instance, if it was sunny and pleasant, it was described as "Hitler weather", or if you failed to comply with Nazi ideals of racial and social conformity, you were "switched off."

When the blatant emphasis on racial hatred of others seemed to reach an impasse in the school system, through radio broadcasts, or on film reels, the overseers of Nazi Gleichschaltung propaganda switched to strategies that focused more on togetherness and the "we-consciousness" of the collective Volk, but the mandates of Nazi "coordination" remained: pay homage to the Führer, expel all foreigners, sacrifice for the German people, and welcome future challenges. While greater German social and economic unity was produced through the Gleichschaltung initiatives of the regime, it was at the expense of individuality and to the social detriment of any nonconformist; and worse—it contributed to and reinforced the social and racial exclusion of anyone deemed an enemy by National Socialist doctrine. The Nazi Gleichschaltung or "synchronization" of German society—along with a series of Nazi legislation—was part and parcel to Jewish economic disenfranchisement, the violence against political opposition, the creation of concentration camps, the Nuremberg Laws, the establishment of a racial Volksgemeinschaft, the seeking of Lebensraum, and the violent mass destruction of human life deemed somehow less valuable by the National Socialist government of Germany.

The Gleichschaltung measures were used to give the Nazi regime the appearance of legality. Thousands of decrees, including many of the Gleichschaltung laws, were based explicitly on the Reichstag Fire Decree, and hence on Article 48. The Reichstag renewed the Enabling Act twice, though this was a foregone conclusion with all other parties banned.

title: Gleichschaltung


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Georg Cantor

Georg Ferdinand Ludwig Philipp Cantor (, ; 3 March 1845 – 6 January 1918 ) was a mathematician who played a pivotal role in the creation of set theory, which has become a fundamental theory in mathematics. Cantor established the importance of one-to-one correspondence between the members of two sets, defined infinite and well-ordered sets, and proved that the real numbers are more numerous than the natural numbers. Cantor's method of proof of this theorem implies the existence of an infinity of infinities. He defined the cardinal and ordinal numbers and their arithmetic. Cantor's work is of great philosophical interest, a fact he was well aware of.

Originally, Cantor's theory of transfinite numbers was regarded as counter-intuitive – even shocking. This caused it to encounter resistance from mathematical contemporaries such as Leopold Kronecker and Henri Poincaré and later from Hermann Weyl and L. E. J. Brouwer, while Ludwig Wittgenstein raised philosophical objections; see Controversy over Cantor's theory. Cantor, a devout Lutheran Christian, believed the theory had been communicated to him by God. Some Christian theologians (particularly neo-Scholastics) saw Cantor's work as a challenge to the uniqueness of the absolute infinity in the nature of God – on one occasion equating the theory of transfinite numbers with pantheism – a proposition that Cantor vigorously rejected. Not all theologians were against Cantor's theory; prominent neo-scholastic philosopher Constantin Gutberlet was in favor of it and Cardinal Johann Baptist Franzelin accepted it as a valid theory (after Cantor made some important clarifications).

The objections to Cantor's work were occasionally fierce: Leopold Kronecker's public opposition and personal attacks included describing Cantor as a "scientific charlatan", a "renegade" and a "corrupter of youth". Kronecker objected to Cantor's proofs that the algebraic numbers are countable, and that the transcendental numbers are uncountable, results now included in a standard mathematics curriculum. Writing decades after Cantor's death, Wittgenstein lamented that mathematics is "ridden through and through with the pernicious idioms of set theory", which he dismissed as "utter nonsense" that is "laughable" and "wrong". Cantor's recurring bouts of depression from 1884 to the end of his life have been blamed on the hostile attitude of many of his contemporaries, though some have explained these episodes as probable manifestations of a bipolar disorder.

The harsh criticism has been matched by later accolades. In 1904, the Royal Society awarded Cantor its Sylvester Medal, the highest honor it can confer for work in mathematics. David Hilbert defended it from its critics by declaring, "No one shall expel us from the paradise that Cantor has created."

Youth and studies
Georg Cantor, born in 1845 in Saint Petersburg, Russian Empire, was brought up in that city until the age of eleven. The oldest of six children, he was regarded as an outstanding violinist. His grandfather Franz Böhm (1788–1846) (the violinist Joseph Böhm's brother) was a well-known musician and soloist in a Russian imperial orchestra. Cantor's father had been a member of the Saint Petersburg stock exchange; when he became ill, the family moved to Germany in 1856, first to Wiesbaden, then to Frankfurt, seeking milder winters than those of Saint Petersburg. In 1860, Cantor graduated with distinction from the Realschule in Darmstadt; his exceptional skills in mathematics, trigonometry in particular, were noted. In August 1862, he then graduated from the "Höhere Gewerbeschule Darmstadt", now the Technische Universität Darmstadt. In 1862 Cantor entered the Swiss Federal Polytechnic in Zurich. After receiving a substantial inheritance upon his father's death in June 1863, Cantor transferred to the University of Berlin, attending lectures by Leopold Kronecker, Karl Weierstrass and Ernst Kummer. He spent the summer of 1866 at the University of Göttingen, then and later a center for mathematical research. Cantor was a good student, and he received his doctoral degree in 1867.

Teacher and researcher
Cantor submitted his dissertation on number theory at the University of Berlin in 1867. After teaching briefly in a Berlin girls' school, he took up a position at the University of Halle, where he spent his entire career. He was awarded the requisite habilitation for his thesis, also on number theory, which he presented in 1869 upon his appointment at Halle University.

In 1874, Cantor married Vally Guttmann. They had six children, the last (Rudolph) born in 1886. Cantor was able to support a family despite his modest academic pay, thanks to his inheritance from his father. During his honeymoon in the Harz mountains, Cantor spent much time in mathematical discussions with Richard Dedekind, whom he had met at Interlaken in Switzerland two years earlier while on holiday.

Cantor was promoted to extraordinary professor in 1872 and made full professor in 1879. To attain the latter rank at the age of 34 was a notable accomplishment, but Cantor desired a chair at a more prestigious university, in particular at Berlin, at that time the leading German university. However, his work encountered too much opposition for that to be possible. Kronecker, who headed mathematics at Berlin until his death in 1891, became increasingly uncomfortable with the prospect of having Cantor as a colleague, perceiving him as a "corrupter of youth" for teaching his ideas to a younger generation of mathematicians. Worse yet, Kronecker, a well-established figure within the mathematical community and Cantor's former professor, disagreed fundamentally with the thrust of Cantor's work ever since he had intentionally delayed the publication of Cantor's first major publication in 1874. Kronecker, now seen as one of the founders of the constructive viewpoint in mathematics, disliked much of Cantor's set theory because it asserted the existence of sets satisfying certain properties, without giving specific examples of sets whose members did indeed satisfy those properties. Whenever Cantor applied for a post in Berlin, he was declined, and the process usually involved Kronecker, so Cantor came to believe that Kronecker's stance would make it impossible for him ever to leave Halle.

In 1881, Cantor's Halle colleague Eduard Heine died. Halle accepted Cantor's suggestion that Heine's vacant chair be offered to Dedekind, Heinrich M. Weber and Franz Mertens, in that order, but each declined the chair after being offered it. Friedrich Wangerin was eventually appointed, but he was never close to Cantor.

In 1882, the mathematical correspondence between Cantor and Dedekind came to an end, apparently as a result of Dedekind's declining the chair at Halle. Cantor also began another important correspondence, with Gösta Mittag-Leffler in Sweden, and soon began to publish in Mittag-Leffler's journal Acta Mathematica. But in 1885, Mittag-Leffler was concerned about the philosophical nature and new terminology in a paper Cantor had submitted to Acta. He asked Cantor to withdraw the paper from Acta while it was in proof, writing that it was "... about one hundred years too soon." Cantor complied, but then curtailed his relationship and correspondence with Mittag-Leffler, writing to a third party, "Had Mittag-Leffler had his way, I should have to wait until the year 1984, which to me seemed too great a demand! ... But of course I never want to know anything again about Acta Mathematica."

Cantor suffered his first known bout of depression in May 1884. Criticism of his work weighed on his mind: every one of the fifty-two letters he wrote to Mittag-Leffler in 1884 mentioned Kronecker. A passage from one of these letters is revealing of the damage to Cantor's self-confidence:

"... I don't know when I shall return to the continuation of my scientific work. At the moment I can do absolutely nothing with it, and limit myself to the most necessary duty of my lectures; how much happier I would be to be scientifically active, if only I had the necessary mental freshness."

This crisis led him to apply to lecture on philosophy rather than on mathematics. He also began an intense study of Elizabethan literature, thinking there might be evidence that Francis Bacon wrote the plays attributed to William Shakespeare (see Shakespearean authorship question); this ultimately resulted in two pamphlets, published in 1896 and 1897.

Cantor recovered soon thereafter, and subsequently made further important contributions, including his diagonal argument and theorem. However, he never again attained the high level of his remarkable papers of 1874–84, even after Kronecker's death on December 29, 1891. He eventually sought, and achieved, a reconciliation with Kronecker. Nevertheless, the philosophical disagreements and difficulties dividing them persisted.

In 1889, Cantor was instrumental in founding the German Mathematical Society, and he chaired its first meeting in Halle in 1891, where he first introduced his diagonal argument; his reputation was strong enough, despite Kronecker's opposition to his work, to ensure he was elected as the first president of this society. Setting aside the animosity Kronecker had displayed towards him, Cantor invited him to address the meeting, but Kronecker was unable to do so because his wife was dying from injuries sustained in a skiing accident at the time. Georg Cantor was also instrumental in the establishment of the first International Congress of Mathematicians, which took place in Zürich, Switzerland, in 1897.

Later years and death
After Cantor's 1884 hospitalization there is no record that he was in any sanatorium again until 1899. Soon after that second hospitalization, Cantor's youngest son Rudolph died suddenly on December 16 (Cantor was delivering a lecture on his views on Baconian theory and William Shakespeare), and this tragedy drained Cantor of much of his passion for mathematics. Cantor was again hospitalized in 1903. One year later, he was outraged and agitated by a paper presented by Julius König at the Third International Congress of Mathematicians. The paper attempted to prove that the basic tenets of transfinite set theory were false. Since the paper had been read in front of his daughters and colleagues, Cantor perceived himself as having been publicly humiliated. Although Ernst Zermelo demonstrated less than a day later that König's proof had failed, Cantor remained shaken, and momentarily questioning God. Cantor suffered from chronic depression for the rest of his life, for which he was excused from teaching on several occasions and repeatedly confined to various sanatoria. The events of 1904 preceded a series of hospitalizations at intervals of two or three years. He did not abandon mathematics completely, however, lecturing on the paradoxes of set theory (Burali-Forti paradox, Cantor's paradox, and Russell's paradox) to a meeting of the Deutsche Mathematiker-Vereinigung in 1903, and attending the International Congress of Mathematicians at Heidelberg in 1904.

In 1911, Cantor was one of the distinguished foreign scholars invited to the 500th anniversary of the founding of the University of St. Andrews in Scotland. Cantor attended, hoping to meet Bertrand Russell, whose newly published Principia Mathematica repeatedly cited Cantor's work, but the encounter did not come about. The following year, St. Andrews awarded Cantor an honorary doctorate, but illness precluded his receiving the degree in person.

Cantor retired in 1913, and lived in poverty and suffered from malnourishment during World War I. The public celebration of his 70th birthday was canceled because of the war. In June 1917, he entered a sanatorium for the last time and continually wrote to his wife asking to be allowed to go home. Georg Cantor had a fatal heart attack on January 6, 1918, in the sanatorium where he had spent the last year of his life.

Mathematical work
Cantor's work between 1874 and 1884 is the origin of set theory. Prior to this work, the concept of a set was a rather elementary one that had been used implicitly since the beginning of mathematics, dating back to the ideas of Aristotle. No one had realized that set theory had any nontrivial content. Before Cantor, there were only finite sets (which are easy to understand) and "the infinite" (which was considered a topic for philosophical, rather than mathematical, discussion). By proving that there are (infinitely) many possible sizes for infinite sets, Cantor established that set theory was not trivial, and it needed to be studied. Set theory has come to play the role of a foundational theory in modern mathematics, in the sense that it interprets propositions about mathematical objects (for example, numbers and functions) from all the traditional areas of mathematics (such as algebra, analysis, and topology) in a single theory, and provides a standard set of axioms to prove or disprove them. The basic concepts of set theory are now used throughout mathematics.

In one of his earliest papers, Cantor proved that the set of real numbers is "more numerous" than the set of natural numbers; this showed, for the first time, that there exist infinite sets of different sizes. He was also the first to appreciate the importance of one-to-one correspondences (hereinafter denoted "1-to-1 correspondence") in set theory. He used this concept to define finite and infinite sets, subdividing the latter into denumerable (or countably infinite) sets and nondenumerable sets (uncountably infinite sets).

Cantor developed important concepts in topology and their relation to cardinality. For example, he showed that the Cantor set, discovered by Henry John Stephen Smith in 1875, is nowhere dense, but has the same cardinality as the set of all real numbers, whereas the rationals are everywhere dense, but countable. He also showed that all countable dense linear orders without end points are order-isomorphic to the rational numbers.

Cantor introduced fundamental constructions in set theory, such as the power set of a set A, which is the set of all possible subsets of A. He later proved that the size of the power set of A is strictly larger than the size of A, even when A is an infinite set; this result soon became known as Cantor's theorem. Cantor developed an entire theory and arithmetic of infinite sets, called cardinals and ordinals, which extended the arithmetic of the natural numbers. His notation for the cardinal numbers was the Hebrew letter $$\aleph$$ (ℵ, aleph) with a natural number subscript; for the ordinals he employed the Greek letter $$\omega$$ (, omega). This notation is still in use today.

The Continuum hypothesis, introduced by Cantor, was presented by David Hilbert as the first of his twenty-three open problems in his address at the 1900 International Congress of Mathematicians in Paris. Cantor's work also attracted favorable notice beyond Hilbert's celebrated encomium. The US philosopher Charles Sanders Peirce praised Cantor's set theory and, following public lectures delivered by Cantor at the first International Congress of Mathematicians, held in Zürich in 1897, Adolf Hurwitz and Jacques Hadamard also both expressed their admiration. At that Congress, Cantor renewed his friendship and correspondence with Dedekind. From 1905, Cantor corresponded with his British admirer and translator Philip Jourdain on the history of set theory and on Cantor's religious ideas. This was later published, as were several of his expository works.

Number theory, trigonometric series and ordinals
Cantor's first ten papers were on number theory, his thesis topic. At the suggestion of Eduard Heine, the Professor at Halle, Cantor turned to analysis. Heine proposed that Cantor solve an open problem that had eluded Peter Gustav Lejeune Dirichlet, Rudolf Lipschitz, Bernhard Riemann, and Heine himself: the uniqueness of the representation of a function by trigonometric series. Cantor solved this problem in 1869. It was while working on this problem that he discovered transfinite ordinals, which occurred as indices n in the nth derived set Sn of a set S of zeros of a trigonometric series. Given a trigonometric series f(x) with S as its set of zeros, Cantor had discovered a procedure that produced another trigonometric series that had S1 as its set of zeros, where S1 is the set of limit points of S. If Sk+1 is the set of limit points of Sk, then he could construct a trigonometric series whose zeros are Sk+1. Because the sets Sk were closed, they contained their limit points, and the intersection of the infinite decreasing sequence of sets S, S1, S2, S3,... formed a limit set, which we would now call Sω, and then he noticed that Sω would also have to have a set of limit points Sω+1, and so on. He had examples that went on forever, and so here was a naturally occurring infinite sequence of infinite numbers ω, ω + 1, ω + 2, ...

Between 1870 and 1872, Cantor published more papers on trigonometric series, and also a paper defining irrational numbers as convergent sequences of rational numbers. Dedekind, whom Cantor befriended in 1872, cited this paper later that year, in the paper where he first set out his celebrated definition of real numbers by Dedekind cuts. While extending the notion of number by means of his revolutionary concept of infinite cardinality, Cantor was paradoxically opposed to theories of infinitesimals of his contemporaries Otto Stolz and Paul du Bois-Reymond, describing them as both "an abomination" and "a cholera bacillus of mathematics". Cantor also published an erroneous "proof" of the inconsistency of infinitesimals.

Set theory
The beginning of set theory as a branch of mathematics is often marked by the publication of Cantor's 1874 paper, "Ueber eine Eigenschaft des Inbegriffes aller reellen algebraischen Zahlen" ("On a Property of the Collection of All Real Algebraic Numbers"). This paper was the first to provide a rigorous proof that there was more than one kind of infinity. Previously, all infinite collections had been implicitly assumed to be equinumerous (that is, of "the same size" or having the same number of elements). Cantor proved that the collection of real numbers and the collection of positive integers are not equinumerous. In other words, the real numbers are not countable. His proof differs from the diagonal argument that he gave in 1891. Cantor's article also contains a new method of constructing transcendental numbers. Transcendental numbers were first constructed by Joseph Liouville in 1844.

Cantor established these results using two constructions. His first construction shows how to write the real algebraic numbers as a sequence a1, a2, a3, .... In other words, the real algebraic numbers are countable. Cantor starts his second construction with any sequence of real numbers. Using this sequence, he constructs nested intervals whose intersection contains a real number not in the sequence. Since every sequence of real numbers can be used to construct a real not in the sequence, the real numbers cannot be written as a sequence – that is, the real numbers are not countable. By applying his construction to the sequence of real algebraic numbers, Cantor produces a transcendental number. Cantor points out that his constructions prove more – namely, they provide a new proof of Liouville's theorem: Every interval contains infinitely many transcendental numbers. Cantor's next article contains a construction that proves the set of transcendental numbers has the same "power" (see below) as the set of real numbers.

Between 1879 and 1884, Cantor published a series of six articles in Mathematische Annalen that together formed an introduction to his set theory. At the same time, there was growing opposition to Cantor's ideas, led by Leopold Kronecker, who admitted mathematical concepts only if they could be constructed in a finite number of steps from the natural numbers, which he took as intuitively given. For Kronecker, Cantor's hierarchy of infinities was inadmissible, since accepting the concept of actual infinity would open the door to paradoxes which would challenge the validity of mathematics as a whole. Cantor also introduced the Cantor set during this period.

The fifth paper in this series, "Grundlagen einer allgemeinen Mannigfaltigkeitslehre" ("Foundations of a General Theory of Aggregates"), published in 1883, was the most important of the six and was also published as a separate monograph. It contained Cantor's reply to his critics and showed how the transfinite numbers were a systematic extension of the natural numbers. It begins by defining well-ordered sets. Ordinal numbers are then introduced as the order types of well-ordered sets. Cantor then defines the addition and multiplication of the cardinal and ordinal numbers. In 1885, Cantor extended his theory of order types so that the ordinal numbers simply became a special case of order types.

In 1891, he published a paper containing his elegant "diagonal argument" for the existence of an uncountable set. He applied the same idea to prove Cantor's theorem: the cardinality of the power set of a set A is strictly larger than the cardinality of A. This established the richness of the hierarchy of infinite sets, and of the cardinal and ordinal arithmetic that Cantor had defined. His argument is fundamental in the solution of the Halting problem and the proof of Gödel's first incompleteness theorem. Cantor wrote on the Goldbach conjecture in 1894.

In 1895 and 1897, Cantor published a two-part paper in Mathematische Annalen under Felix Klein's editorship; these were his last significant papers on set theory. The first paper begins by defining set, subset, etc., in ways that would be largely acceptable now. The cardinal and ordinal arithmetic are reviewed. Cantor wanted the second paper to include a proof of the continuum hypothesis, but had to settle for expositing his theory of well-ordered sets and ordinal numbers. Cantor attempts to prove that if A and B are sets with A equivalent to a subset of B and B equivalent to a subset of A, then A and B are equivalent. Ernst Schröder had stated this theorem a bit earlier, but his proof, as well as Cantor's, was flawed. Felix Bernstein supplied a correct proof in his 1898 PhD thesis; hence the name Cantor–Bernstein–Schröder theorem.

One-to-one correspondence
Cantor's 1874 Crelle paper was the first to invoke the notion of a 1-to-1 correspondence, though he did not use that phrase. He then began looking for a 1-to-1 correspondence between the points of the unit square and the points of a unit line segment. In an 1877 letter to Richard Dedekind, Cantor proved a far stronger result: for any positive integer n, there exists a 1-to-1 correspondence between the points on the unit line segment and all of the points in an n-dimensional space. About this discovery Cantor wrote to Dedekind: "Je le vois, mais je ne le crois pas!" ("I see it, but I don't believe it!") The result that he found so astonishing has implications for geometry and the notion of dimension.

In 1878, Cantor submitted another paper to Crelle's Journal, in which he defined precisely the concept of a 1-to-1 correspondence and introduced the notion of "power" (a term he took from Jakob Steiner) or "equivalence" of sets: two sets are equivalent (have the same power) if there exists a 1-to-1 correspondence between them. Cantor defined countable sets (or denumerable sets) as sets which can be put into a 1-to-1 correspondence with the natural numbers, and proved that the rational numbers are denumerable. He also proved that n-dimensional Euclidean space Rn has the same power as the real numbers R, as does a countably infinite product of copies of R. While he made free use of countability as a concept, he did not write the word "countable" until 1883. Cantor also discussed his thinking about dimension, stressing that his mapping between the unit interval and the unit square was not a continuous one.

This paper displeased Kronecker and Cantor wanted to withdraw it; however, Dedekind persuaded him not to do so and Karl Weierstrass supported its publication. Nevertheless, Cantor never again submitted anything to Crelle.

Continuum hypothesis
Cantor was the first to formulate what later came to be known as the continuum hypothesis or CH: there exists no set whose power is greater than that of the naturals and less than that of the reals (or equivalently, the cardinality of the reals is exactly aleph-one, rather than just at least aleph-one). Cantor believed the continuum hypothesis to be true and tried for many years to prove it, in vain. His inability to prove the continuum hypothesis caused him considerable anxiety.

The difficulty Cantor had in proving the continuum hypothesis has been underscored by later developments in the field of mathematics: a 1940 result by Kurt Gödel and a 1963 one by Paul Cohen together imply that the continuum hypothesis can be neither proved nor disproved using standard Zermelo–Fraenkel set theory plus the axiom of choice (the combination referred to as "ZFC").

Absolute infinite, well-ordering theorem, and paradoxes
In 1883, Cantor divided the infinite into the transfinite and the absolute.

The transfinite is increasable in magnitude, while the absolute is unincreasable. For example, an ordinal α is transfinite because it can be increased to α + 1. On the other hand, the ordinals form an absolutely infinite sequence that cannot be increased in magnitude because there are no larger ordinals to add to it. In 1883, Cantor also introduced the well-ordering principle "every set can be well-ordered" and stated that it is a "law of thought".

Cantor extended his work on the absolute infinite by using it in a proof. Around 1895, he began to regard his well-ordering principle as a theorem and attempted to prove it. In 1899, he sent Dedekind a proof of the equivalent aleph theorem: the cardinality of every infinite set is an aleph. First, he defined two types of multiplicities: consistent multiplicities (sets) and inconsistent multiplicities (absolutely infinite multiplicities). Next he assumed that the ordinals form a set, proved that this leads to a contradiction, and concluded that the ordinals form an inconsistent multiplicity. He used this inconsistent multiplicity to prove the aleph theorem. In 1932, Zermelo criticized the construction in Cantor's proof.

Cantor avoided paradoxes by recognizing that there are two types of multiplicities. In his set theory, when it is assumed that the ordinals form a set, the resulting contradiction implies only that the ordinals form an inconsistent multiplicity. In contrast, Bertrand Russell treated all collections as sets, which leads to paradoxes. In Russell's set theory, the ordinals form a set, so the resulting contradiction implies that the theory is inconsistent. From 1901 to 1903, Russell discovered three paradoxes implying that his set theory is inconsistent: the Burali-Forti paradox (which was just mentioned), Cantor's paradox, and Russell's paradox. Russell named paradoxes after Cesare Burali-Forti and Cantor even though neither of them believed that they had found paradoxes.

In 1908, Zermelo published his axiom system for set theory. He had two motivations for developing the axiom system: eliminating the paradoxes and securing his proof of the well-ordering theorem. Zermelo had proved this theorem in 1904 using the axiom of choice, but his proof was criticized for a variety of reasons. His response to the criticism included his axiom system and a new proof of the well-ordering theorem. His axioms support this new proof, and they eliminate the paradoxes by restricting the formation of sets.

In 1923, John von Neumann developed an axiom system that eliminates the paradoxes by using an approach similar to Cantor's—namely, by identifying collections that are not sets and treating them differently. Von Neumann stated that a class is too big to be a set if it can be put into one-to-one correspondence with the class of all sets. He defined a set as a class that is a member of some class and stated the axiom: A class is not a set if and only if there is a one-to-one correspondence between it and the class of all sets. This axiom implies that these big classes are not sets, which eliminates the paradoxes since they cannot be members of any class. Von Neumann also used his axiom to prove the well-ordering theorem: Like Cantor, he assumed that the ordinals form a set. The resulting contradiction implies that the class of all ordinals is not a set. Then his axiom provides a one-to-one correspondence between this class and the class of all sets. This correspondence well-orders the class of all sets, which implies the well-ordering theorem. In 1930, Zermelo defined models of set theory that satisfy von Neumann's axiom.

Philosophy, religion, literature and Cantor's mathematics
The concept of the existence of an actual infinity was an important shared concern within the realms of mathematics, philosophy and religion. Preserving the orthodoxy of the relationship between God and mathematics, although not in the same form as held by his critics, was long a concern of Cantor's. He directly addressed this intersection between these disciplines in the introduction to his Grundlagen einer allgemeinen Mannigfaltigkeitslehre, where he stressed the connection between his view of the infinite and the philosophical one. To Cantor, his mathematical views were intrinsically linked to their philosophical and theological implications – he identified the absolute infinite with God, and he considered his work on transfinite numbers to have been directly communicated to him by God, who had chosen Cantor to reveal them to the world. He was a devout Lutheran whose explicit Christian beliefs shaped his philosophy of science. Joseph Dauben has traced the effect Cantor's Christian convictions had on the development of transfinite set theory.

Debate among mathematicians grew out of opposing views in the philosophy of mathematics regarding the nature of actual infinity. Some held to the view that infinity was an abstraction which was not mathematically legitimate, and denied its existence. Mathematicians from three major schools of thought (constructivism and its two offshoots, intuitionism and finitism) opposed Cantor's theories in this matter. For constructivists such as Kronecker, this rejection of actual infinity stems from fundamental disagreement with the idea that nonconstructive proofs such as Cantor's diagonal argument are sufficient proof that something exists, holding instead that constructive proofs are required. Intuitionism also rejects the idea that actual infinity is an expression of any sort of reality, but arrive at the decision via a different route than constructivism. Firstly, Cantor's argument rests on logic to prove the existence of transfinite numbers as an actual mathematical entity, whereas intuitionists hold that mathematical entities cannot be reduced to logical propositions, originating instead in the intuitions of the mind. Secondly, the notion of infinity as an expression of reality is itself disallowed in intuitionism, since the human mind cannot intuitively construct an infinite set. Mathematicians such as L. E. J. Brouwer and especially Henri Poincaré adopted an intuitionist stance against Cantor's work. Finally, Wittgenstein's attacks were finitist: he believed that Cantor's diagonal argument conflated the intension of a set of cardinal or real numbers with its extension, thus conflating the concept of rules for generating a set with an actual set.

Some Christian theologians saw Cantor's work as a challenge to the uniqueness of the absolute infinity in the nature of God. In particular, neo-Thomist thinkers saw the existence of an actual infinity that consisted of something other than God as jeopardizing "God's exclusive claim to supreme infinity". Cantor strongly believed that this view was a misinterpretation of infinity, and was convinced that set theory could help correct this mistake: "... the transfinite species are just as much at the disposal of the intentions of the Creator and His absolute boundless will as are the finite numbers.". Prominent neo-scholastic German philosopher Constantin Gutberlet was in favor of such theory, holding that it didn't oppose the nature of God.

Cantor also believed that his theory of transfinite numbers ran counter to both materialism and determinism – and was shocked when he realized that he was the only faculty member at Halle who did not hold to deterministic philosophical beliefs.

It was important to Cantor that his philosophy provided an "organic explanation" of nature, and in his 1883 Grundlagen, he said that such an explanation could only come about by drawing on the resources of the philosophy of Spinoza and Leibniz. In making these claims, Cantor may have been influenced by F. A. Trendelenburg, whose lecture courses he attended at Berlin, and in turn Cantor produced a Latin commentary on Book 1 of Spinoza's Ethica. Trendelenburg was also the examiner of Cantor's Habilitationsschrift.

In 1888, Cantor published his correspondence with several philosophers on the philosophical implications of his set theory. In an extensive attempt to persuade other Christian thinkers and authorities to adopt his views, Cantor had corresponded with Christian philosophers such as Tilman Pesch and Joseph Hontheim, as well as theologians such as Cardinal Johann Baptist Franzelin, who once replied by equating the theory of transfinite numbers with pantheism. Although later this Cardinal accepted the theory as valid, due to some clarifications from Cantor's. Cantor even sent one letter directly to Pope Leo XIII himself, and addressed several pamphlets to him.

Cantor's philosophy on the nature of numbers led him to affirm a belief in the freedom of mathematics to posit and prove concepts apart from the realm of physical phenomena, as expressions within an internal reality. The only restrictions on this metaphysical system are that all mathematical concepts must be devoid of internal contradiction, and that they follow from existing definitions, axioms, and theorems. This belief is summarized in his assertion that "the essence of mathematics is its freedom." These ideas parallel those of Edmund Husserl, whom Cantor had met in Halle.

Meanwhile, Cantor himself was fiercely opposed to infinitesimals, describing them as both an "abomination" and "the cholera bacillus of mathematics".

Cantor's 1883 paper reveals that he was well aware of the opposition his ideas were encountering: "... I realize that in this undertaking I place myself in a certain opposition to views widely held concerning the mathematical infinite and to opinions frequently defended on the nature of numbers."

Hence he devotes much space to justifying his earlier work, asserting that mathematical concepts may be freely introduced as long as they are free of contradiction and defined in terms of previously accepted concepts. He also cites Aristotle, René Descartes, George Berkeley, Gottfried Leibniz, and Bernard Bolzano on infinity. Instead, he always strongly rejected Immanuel Kant's philosophy, in the realms of both the philosophy of mathematics and metaphysics. He shared B. Russell's motto "Kant or Cantor", and defined Kant "yonder sophistical Philistine who knew so little mathematics."

Cantor's ancestry
Cantor's paternal grandparents were from Copenhagen and fled to Russia from the disruption of the Napoleonic Wars. There is very little direct information on them. Cantor's father, Georg Waldemar Cantor, was educated in the Lutheran mission in Saint Petersburg, and his correspondence with his son shows both of them as devout Lutherans. Very little is known for sure about Georg Waldemar's origin or education. Cantor's mother, Maria Anna Böhm, was an Austro-Hungarian born in Saint Petersburg and baptized Roman Catholic; she converted to Protestantism upon marriage. However, there is a letter from Cantor's brother Louis to their mother, stating: "Mögen wir zehnmal von Juden abstammen und ich im Princip noch so sehr für Gleichberechtigung der Hebräer sein, im socialen Leben sind mir Christen lieber ..."

("Even if we were descended from Jews ten times over, and even though I may be, in principle, completely in favour of equal rights for Hebrews, in social life I prefer Christians...") which could be read to imply that she was of Jewish ancestry.

According to biographers Eric Temple Bell, Cantor was of Jewish descent, although both parents were baptized. In a 1971 article entitled "Towards a Biography of Georg Cantor", the British historian of mathematics Ivor Grattan-Guinness mentions (Annals of Science 27, pp. 345–391, 1971) that he was unable to find evidence of Jewish ancestry. (He also states that Cantor's wife, Vally Guttmann, was Jewish).

In a letter written to Paul Tannery in 1896 (Paul Tannery, Memoires Scientifique 13 Correspondence, Gauthier-Villars, Paris, 1934, p. 306), Cantor states that his paternal grandparents were members of the Sephardic Jewish community of Copenhagen. Specifically, Cantor states in describing his father: "Er ist aber in Kopenhagen geboren, von israelitischen Eltern, die der dortigen portugisischen Judengemeinde...." ("He was born in Copenhagen of Jewish (lit: 'Israelite') parents from the local Portuguese-Jewish community.") In addition, Cantor's maternal great uncle, a Hungarian violinist Josef Böhm, has been described as Jewish, which may imply that Cantor's mother was at least partly descended from the Hungarian Jewish community.

In a letter to Bertrand Russell, Cantor described his ancestry and self-perception as follows: "Neither my father nor my mother were of German blood, the first being a Dane, borne in Kopenhagen, my mother of Austrian Hungar descension. You must know, Sir, that I am not a regular just Germain, for I am born 3 March 1845 at Saint Peterborough, Capital of Russia, but I went with my father and mother and brothers and sister, eleven years old in the year 1856, into Germany." There were documented statements, during the 1930s, that called this Jewish ancestry into question: "More often [i.e., than the ancestry of the mother] the question has been discussed of whether Georg Cantor was of Jewish origin. About this it is reported in a notice of the Danish genealogical Institute in Copenhagen from the year 1937 concerning his father: 'It is hereby testified that Georg Woldemar Cantor, born 1809 or 1814, is not present in the registers of the Jewish community, and that he completely without doubt was not a Jew ...'"

Biographies
Until the 1970s, the chief academic publications on Cantor were two short monographs by Arthur Moritz Schönflies (1927) – largely the correspondence with Mittag-Leffler – and Fraenkel (1930). Both were at second and third hand; neither had much on his personal life. The gap was largely filled by Eric Temple Bell's Men of Mathematics (1937), which one of Cantor's modern biographers describes as "perhaps the most widely read modern book on the history of mathematics"; and as "one of the worst". Bell presents Cantor's relationship with his father as Oedipal, Cantor's differences with Kronecker as a quarrel between two Jews, and Cantor's madness as Romantic despair over his failure to win acceptance for his mathematics. Grattan-Guinness (1971) found that none of these claims were true, but they may be found in many books of the intervening period, owing to the absence of any other narrative. There are other legends, independent of Bell – including one that labels Cantor's father a foundling, shipped to Saint Petersburg by unknown parents. A critique of Bell's book is contained in Joseph Dauben's biography. Writes Dauben: "Cantor devoted some of his most vituperative correspondence, as well as a portion of the Beiträge, to attacking what he described at one point as the 'infinitesimal Cholera bacillus of mathematics', which had spread from Germany through the work of Thomae, du Bois Reymond and Stolz, to infect Italian mathematics ... Any acceptance of infinitesimals necessarily meant that his own theory of number was incomplete. Thus to accept the work of Thomae, du Bois-Reymond, Stolz and Veronese was to deny the perfection of Cantor's own creation. Understandably, Cantor launched a thorough campaign to discredit Veronese's work in every way possible."

Primary literature in German

 * Published separately as: Grundlagen einer allgemeinen Mannigfaltigkeitslehre.
 * . Almost everything that Cantor wrote. Includes excerpts of his correspondence with Dedekind (p. 443–451) and Fraenkel's Cantor biography (p. 452–483) in the appendix.
 * Published separately as: Grundlagen einer allgemeinen Mannigfaltigkeitslehre.
 * . Almost everything that Cantor wrote. Includes excerpts of his correspondence with Dedekind (p. 443–451) and Fraenkel's Cantor biography (p. 452–483) in the appendix.
 * Published separately as: Grundlagen einer allgemeinen Mannigfaltigkeitslehre.
 * . Almost everything that Cantor wrote. Includes excerpts of his correspondence with Dedekind (p. 443–451) and Fraenkel's Cantor biography (p. 452–483) in the appendix.
 * Published separately as: Grundlagen einer allgemeinen Mannigfaltigkeitslehre.
 * . Almost everything that Cantor wrote. Includes excerpts of his correspondence with Dedekind (p. 443–451) and Fraenkel's Cantor biography (p. 452–483) in the appendix.
 * . Almost everything that Cantor wrote. Includes excerpts of his correspondence with Dedekind (p. 443–451) and Fraenkel's Cantor biography (p. 452–483) in the appendix.
 * . Almost everything that Cantor wrote. Includes excerpts of his correspondence with Dedekind (p. 443–451) and Fraenkel's Cantor biography (p. 452–483) in the appendix.
 * . Almost everything that Cantor wrote. Includes excerpts of his correspondence with Dedekind (p. 443–451) and Fraenkel's Cantor biography (p. 452–483) in the appendix.
 * . Almost everything that Cantor wrote. Includes excerpts of his correspondence with Dedekind (p. 443–451) and Fraenkel's Cantor biography (p. 452–483) in the appendix.

Secondary literature

 * . ISBN 0-7607-7778-0. A popular treatment of infinity, in which Cantor is frequently mentioned.
 * . ISBN 3-7643-8349-6 Contains a detailed treatment of both Cantor's and Dedekind's contributions to set theory.
 * . ISBN 3-540-90092-6
 * . ISBN 0-8126-9538-0 Three chapters and 18 index entries on Cantor.
 * Newstead, Anne (2009). "Cantor on Infinity in Nature, Number, and the Divine Mind", American Catholic Philosophical Quarterly, 83 (4): 532–553, https://doi.org/10.5840/acpq200983444. With acknowledgement of Dauben's pioneering historical work, this article further discusses Cantor's relation to the philosophy of Spinoza and Leibniz in depth, and his engagement in the Pantheismusstreit. Brief mention is made of Cantor's learning from F.A.Trendelenburg.
 * . ISBN 0-679-77631-1 Chapter 16 illustrates how Cantorian thinking intrigues a leading contemporary theoretical physicist.
 * . ISBN 0-553-25531-2 Deals with similar topics to Aczel, but in more depth.
 * Leonida Lazzari, L'infinito di Cantor. Editrice Pitagora, Bologna, 2008.
 * . ISBN 0-679-77631-1 Chapter 16 illustrates how Cantorian thinking intrigues a leading contemporary theoretical physicist.
 * . ISBN 0-553-25531-2 Deals with similar topics to Aczel, but in more depth.
 * Leonida Lazzari, L'infinito di Cantor. Editrice Pitagora, Bologna, 2008.
 * Leonida Lazzari, L'infinito di Cantor. Editrice Pitagora, Bologna, 2008.

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title: Georg Cantor


text:
University of Gothenburg

The University of Gothenburg (Göteborgs universitet) is a university in Sweden's second largest city, Gothenburg. Founded in 1891, the university is the third-oldest of the current Swedish universities and, with 53,624 students and 6,707 staff members, it is one of the largest universities in the Nordic countries.

About
With its eight faculties and 38 departments, the University of Gothenburg is one of the most wide-ranging and versatile universities in Sweden. Its eight faculties offer training in the Creative Arts, Social Sciences, Natural Sciences, Humanities, Education, Information Technology, Business, Economics and Law, and Health Sciences.

The University of Gothenburg has the highest number of applicants per study place in many of its subjects and courses, making it one of the most popular universities in Sweden.

History
The University of Gothenburg was founded as Göteborgs högskola (Gothenburg University College) in 1891. In 1907 it was granted the same status as Uppsala University and Lund University by the Swedish government, thus creating Sweden's third university.

Over the course of time, it has merged with a number of previously independent higher education institutions in the city and has continuously expanded its study profile. It was granted the rights of a full university by the Swedish Government in 1954, following the merger of the Göteborgs högskola (Gothenburg College) with the Medicinhögskolan i Göteborg (Gothenburg Medical School).

The University of Gothenburg is a pronounced city university, that is most of its facilities are within the city centre of Gothenburg. The main building as well as most faculties are located in the central part of Gothenburg.

Together with Uppsala, Lund, and Stockholm universities, it is one of four large international research universities in Sweden.

Management
The University of Gothenburg is one of Sweden's largest universities. It is a comprehensive university, organised into eight faculties and 38 departments. The university is a public authority as well as a confederation of Faculty Boards. Each faculty/school has significant autonomy based on its attributed powers, and a distinct identity within the university.

The University Board is the university's highest decision-making body. The board consists has "supervision over all the University's affairs, and is responsible that its duties are fulfilled". The Swedish Government appoints seven of the members externally, based on their having experience in activities that have significance for the university's teaching and research functions. In addition, the vice-chancellor, three faculty members and three students, as well as union representatives are included as ordinary members.

The day-to-day management is headed by the vice-chancellor, Malin Broberg, who is responsible for implementing the decisions of the board. She is supported by the central administration.

Faculties
The university is organised into eight academic faculties. Collaboration across faculty and subject boundaries is emphasised in the university's research and education strategies. All faculties takes advantage of this possibility and are active participants in a multitude of cross-disciplinary research and education activities within the framework of the university. The university closely cooperates with Chalmers University of Technology, a fact that further increases the total scope of the academic environment in Gothenburg.
 * The Artistic Faculty (Konstnärliga fakulteten) offers courses in the fields of design and crafts, film studies, photography, scene, music and fine arts.
 * The Faculty of Education (Utbildningsvetenskapliga fakulteten) is responsible for teacher training in various subjects.
 * The Faculty of Humanities (Humanistiska fakulteten) comprises the humanities, for instance cultural studies, history, literature, history of ideas, religion, archeology, modern languages, philosophy, linguistics, theory of science and Swedish language and literature.
 * The IT Faculty (IT fakulteten) offers programmes in applied information technology, computer science and engineering.
 * The Faculty of Natural Sciences (Naturvetenskapliga fakulteten) covers a number of science disciplines such as botany, cell and molecular biology, physics, earth sciences, chemistry, conservation, marine ecology, mathematics, environmental science, and zoology.
 * The Sahlgrenska Academy (Sahlgrenska Akademin) is the faculty of medicine at the University of Gothenburg. Research and education in medicine, odontology and health and care sciences.
 * The School of Business, Economics and Law (Handelshögskolan vid Göteborgs universitet) is a business school and a law school.
 * The Faculty of Social Sciences (Samhällsvetenskapliga fakulteten) offers courses in peace and development studies, public administration, journalism, psychology, social anthropology, social work, sociology, political science, and European studies.

World ranking
University of Gothenburg was ranked number 201 in the world by Times Higher Educations 2023.

Alumni

 * Sture Allén, computer linguist, former permanent secretary of the Swedish Academy
 * Jens Allwood, linguist
 * Muhammad al-Yaqoubi, Syrian Islamic scholar
 * Dilruba Z. Ara, writer
 * Percy Barnevik, industry leader, former CEO of Asea Brown Boveri
 * Nick Bostrom, philosopher and futurist, and Director of the Future of Humanity Institute at Oxford University
 * Malin Byström, operatic soprano
 * Arvid Carlsson, Nobel Prize laureate in Medicine, 2000
 * Magnus Carlsson, singer
 * Ernst Cassirer, philosopher
 * Fredy Clue, artist and musician
 * Östen Dahl, linguist (member of the Royal Swedish Academy of Sciences, Royal Swedish Academy of Letters, History and Antiquities and Academia Europaea)
 * Åke Edwardson, author (used to teach Journalism)
 * Jan Eliasson, diplomat and politician (former President of the United Nations General Assembly, former Minister of Foreign Affairs of the Kingdom of Sweden)
 * Carl Henrik Fredriksson, literary critic, columnist, essayist, and translator
 * José González, singer
 * Aditya K. Gupta, Canadian medical academic, dermatologist and medical mycology researcher
 * Hans Hagnell, politician
 * Gunnar D Hansson, author
 * Heiða Björg Hilmisdóttir, Icelandic politician
 * Tore Janson, linguist.
 * Ann-Sofie Johansson, Creative Advisor and former Head of Design for H&M
 * Jonas Jonasson, author of The Hundred-Year-Old Man Who Climbed Out the Window and Disappeared.
 * Bernhard Karlgren, sinologist
 * Nils Kock, developer of the Kock pouch surgical procedure
 * Juri Kurol, Swedish Orthodontist
 * Erik Lönnroth, historian (Member of the Swedish Academy)
 * Lotta Lotass, writer and literary scholar (former Member of the Swedish Academy)
 * Cecilia Malmström, former European Commissioner
 * Aleksandra Mir, visual artist
 * Njuguna Ndungu, economist, Governor and Chairman of the Board of the Central Bank of Kenya
 * Joakim Norbeck, scientist in molecular biotechnology at Chalmers University
 * Bo Ralph, linguist (Member of the Swedish Academy)
 * Bo Rothstein, political scientist
 * Susanna Roxman, Anglophone poet and critic
 * Gudrun Slettengren-Fernholm, artist
 * Anna Wåhlin, Antarctic researcher, oceanographer
 * Martin Wallström, actor
 * Maria Wetterstrand, politician (Green Party)
 * Agnes Wold, professor of clinical bacteriology

Vice-Chancellors of the University of Gothenburg

 * 1891 Axel Kock
 * 1891–1893 Hjalmar Edgren
 * 1893–1899 Johannes Paulson
 * 1899–1909 Johan Vising
 * 1909–1914 Ludvig Stavenow
 * 1914–1931 Otto Sylwan
 * 1931–1936 Bernhard Karlgren
 * 1936–1951 Curt Weibull
 * 1951–1966 Hjalmar Frisk
 * 1966–1972 Bo Eric Ingelmark
 * 1972–1982 Georg Lundgren
 * 1982–1986 Kjell Härnqvist
 * 1986–1992 Jan S. Nilsson
 * 1992–1997 Jan Ling
 * 1997–2003 Bo Samuelsson
 * 2003–2006 Gunnar Svedberg
 * 2006–2017 Pam Fredman
 * 2017–2023 Eva Wiberg
 * 1st of July 2023 Malin Broberg

title: University of Gothenburg


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Film genre

A film genre is a stylistic or thematic category for motion pictures based on similarities either in the narrative elements, aesthetic approach, or the emotional response to the film.

Drawing heavily from the theories of literary-genre criticism, film genres are usually delineated by "conventions, iconography, settings, narratives, characters and actors". One can also classify films by the tone, theme/topic, mood, format, target audience, or budget. These characteristics are most evident in genre films, which are "commercial feature films [that], through repetition and variation, tell familiar stories with familiar characters and familiar situations" in a given genre.

A film's genre will influence the use of filmmaking styles and techniques, such as the use of flashbacks and low-key lighting in film noir; tight framing in horror films; or fonts that look like rough-hewn logs for the titles of Western films. In addition, genres have associated film scoring conventions, such as lush string orchestras for romantic melodramas or electronic music for science fiction films. Genre also affects how films are broadcast on television, advertised, and organized in video rental stores.

Alan Williams distinguishes three main genre categories: narrative, avant-garde, and documentary.

With the proliferation of particular genres, film subgenres can also emerge: the legal drama, for example, is a sub-genre of drama that includes courtroom- and trial-focused films. Subgenres are often a mixture of two separate genres; genres can also merge with seemingly unrelated ones to form hybrid genres, where popular combinations include the romantic comedy and the action comedy film. Broader examples include the docufiction and docudrama, which merge the basic categories of fiction and non-fiction (documentary).

Genres are not fixed; they change and evolve over time, and some genres may largely disappear (for example, the melodrama). Not only does genre refer to a type of film or its category, a key role is also played by the expectations of an audience about a film, as well as institutional discourses that create generic structures.

Characteristics
Characteristics of particular genres are most evident in genre films, which are "commercial feature films [that], through repetition and variation, tell familiar stories with familiar characters and familiar situations" in a given genre.

Drawing heavily from the theories of literary-genre criticism, film genres are usually delineated by conventions, iconography, narratives, formats, characters, and actors, all of which can vary according to the genre. In terms of standard or "stock" characters, those in film noir, for example, include the femme fatale and the "hardboiled" detective; while those in Westerns, stock characters include the schoolmarm and the gunslinger. Regarding actors, some may acquire a reputation linked to a single genre, such as John Wayne (the Western) or Fred Astaire (the musical). Some genres have been characterized or known to use particular formats, which refers to the way in which films are shot (e.g., 35 mm, 16 mm or 8 mm) or the manner of presentation (e.g., anamorphic widescreen).

Genres can also be classified by more inherent characteristics (usually implied in their names), such as settings, theme/topic, mood, target audience, or budget/type of production.

 * The setting is the environment—including both time and geographic location—in which the story and action take place (e.g., present day or historical period; Earth or outer-space; urban or rural, etc.). Genres that are particularly concerned with this element include the historical drama, war film, Western, and space-opera, the names of which all denote particular settings.
 * The theme or topic refers to the issues or concepts that the film revolves around; for example, the science fiction film, sports film, and crime film.
 * The mood is the emotional tone of the film, as implied in the names of the comedy film, horror film, or 'tearjerker'.
 * Genres informed by particular target audience(s) include children's film, teen film, woman's film, and "chick flick".
 * Genres characterized by the type of production include the blockbuster, independent film, and low-budget film, such as the B movie (commercial) or amateur film (noncommercial).

Screenwriters, in particular, often organize their stories by genre, focusing their attention on three specific aspects: atmosphere, character, and story. A film's atmosphere includes costumes, props, locations, and the visceral experiences created for the audience. Aspects of character include archetypes, stock characters, and the goals and motivations of the central characters. Some story considerations for screenwriters, as they relate to genre, include theme, tent-pole scenes, and how the rhythm of characters' perspective shift from scene to scene.

History
From the earliest days of cinema in the 19th century the term "genre" (already in use in English with reference to works of art or literary production from at least 1770 ) was used to organize films according to type. By the 1950s André Bazin was discussing the concept of "genre" by using the Western film as an example; during this era, there was a debate over auteur theory versus genre. In the late 1960s the concept of genre became a significant part of film theory.

Film genres draw on genres from other forms; Western novels existed before the Western film, and musical theatre pre-dated film musicals. The perceived genre of a film can change over time; for example, in the 21st century The Great Train Robbery (1903) classes as a key early Western film, but when released, marketing promoted it "for its relation to the then-popular genres of the chase film, the railroad film and the crime film". A key reason that the early Hollywood industrial system from the 1920s to the 1950s favoured genre films is that in "Hollywood's industrial mode of production, genre movies are dependable products" to market to audiences – they were easy to produce and it was easy for audiences to understand a genre film. In the 1920s to 1950s, genre films had clear conventions and iconography, such as the heavy coats worn by gangsters in films like Little Caesar (1931). The conventions in genre films enable filmmakers to generate them in an industrial, assembly-line fashion, an approach which can be seen in the James Bond spy-films, which all use a formula of "lots of action, fancy gadgets, beautiful woman and colourful villains", even though the actors, directors and screenwriters change.

Pure and hybrid genres
Films are rarely purely from one genre, which is in keeping with the cinema's diverse and derivative origins, it being a blend of "vaudeville, music-hall, theatre, photography" and novels. American film historian Janet Staiger states that the genre of a film can be defined in four ways. The "idealist method" judges films by predetermined standards. The "empirical method" identifies the genre of a film by comparing it to a list of films already deemed to fall within a certain genre. The a priori method uses common generic elements which are identified in advance. The "social conventions" method of identifying the genre of a film is based on the accepted cultural consensus within society. Martin Loop contends that Hollywood films are not pure genres because most Hollywood movies blend the love-oriented plot of the romance genre with other genres. Jim Colins claims that since the 1980s, Hollywood films have been influenced by the trend towards "ironic hybridization", in which directors combine elements from different genres, as with the Western/science fiction mix in Back to the Future Part III.

Many films cross into multiple genres. Susan Hayward states that spy films often cross genre boundaries with thriller films. Some genre films take genre elements from one genre and place them into the conventions of a second genre, such as with The Band Wagon (1953), which adds film noir and detective film elements into "The Girl Hunt" ballet. In the 1970s New Hollywood era, there was so much parodying of genres that it can be hard to assign genres to some films from this era, such as Mel Brooks' comedy-Western Blazing Saddles (1974) or the private eye parody The Long Goodbye (1973). Other films from this era bend genres so much that it is challenging to put them in a genre category, such as Roman Polanski's Chinatown (1974) and William Friedkin's The French Connection (1971).

Film theorist Robert Stam challenged whether genres really exist, or whether they are merely made up by critics. Stam has questioned whether "genres [are] really 'out there' in the world or are they really the construction of analysts?". As well, he has asked whether there is a "... finite taxonomy of genres or are they in principle infinite?" and whether genres are "...timeless essences ephemeral, time-bound entities? Are genres culture-bound or trans-cultural?". Stam has also asked whether genre analysis should aim at being descriptive or prescriptive. While some genres are based on story content (the war film), other are borrowed from literature (comedy, melodrama) or from other media (the musical). Some are performer-based (Fred Astaire and Ginger Rogers films) or budget-based (blockbusters, low-budget film), while others are based on artistic status (the art film), racial identity (race films), location (the Western), or sexual orientation ("New Queer Cinema").

Audience expectations
Many genres have built-in audiences and corresponding publications that support them, such as magazines and websites. For example, horror films have a well-established fanbase that reads horror magazines such as Fangoria. Films that are difficult to categorize into a genre are often less successful. As such, film genres are also useful in the areas of marketing, film criticism and the analysis of consumption. Hollywood story consultant John Truby states that "...you have to know how to transcend the forms [genres] so you can give the audience a sense of originality and surprise".

Some screenwriters use genre as a means of determining what kind of plot or content to put into a screenplay. They may study films of specific genres to find examples. This is a way that some screenwriters are able to copy elements of successful movies and pass them off in a new screenplay. It is likely that such screenplays fall short in originality. As Truby says, "Writers know enough to write a genre script but they haven't twisted the story beats of that genre in such a way that it gives an original face to it".

Cinema technologies are associated with genres. Huge widescreens helped Western films to create an expansive setting of the open plains and desert. Science fiction and fantasy films are associated with special effects, notably computer generated imagery (e.g., the Harry Potter films).

In 2017, screenwriter Eric R. Williams published a system for screenwriters to conceptualize narrative film genres based on audience expectations. The system was based upon the structure biologists use to analyze living beings. Williams wrote a companion book detailing his taxonomy, which claims to be able to identify all feature length narrative films with seven categorizations: film type, super genre, macro-genre, micro-genre, voice, and pathway.

Categorization
Because genres are easier to recognize than to define, academics agree they cannot be identified in a rigid way. Furthermore, different countries and cultures define genres in different ways. A typical example are war movies. In United States, they are mostly related to ones with large U.S. involvement such as World wars and Vietnam, whereas in other countries, movies related to wars in other historical periods are considered war movies.

Film genres may appear to be readily categorizable from the setting of the film. Nevertheless, films with the same settings can be very different, due to the use of different themes or moods. For example, while both The Battle of Midway and All Quiet on the Western Front are set in a wartime context and might be classified as belonging to the war film genre, the first examines the themes of honor, sacrifice, and valour, and the second is an anti-war film which emphasizes the pain and horror of war. While there is an argument that film noir movies could be deemed to be set in an urban setting, in cheap hotels and underworld bars, many classic noirs take place mainly in small towns, suburbia, rural areas, or on the open road.

The editors of filmsite.org argue that animation, pornographic film, documentary film, silent film and so on are non-genre-based film categories.

Linda Williams argues that horror, melodrama, and pornography all fall into the category of "body genres" since they are each designed to elicit physical reactions on the part of viewers. Horror is designed to elicit spine-chilling, white-knuckled, eye-bulging terror; melodramas are designed to make viewers cry after seeing the misfortunes of the onscreen characters; and pornography is designed to elicit sexual arousal. This approach can be extended: comedies make people laugh, tear-jerkers make people cry, feel-good films lift people's spirits and inspiration films provide hope for viewers.

Eric R. Williams (no relation to Linda Williams) argues that all narrative feature-length films can be categorized as one of eleven "super genres" (action, crime, fantasy, horror, romance, science fiction, slice of life, sports, thriller, war and Western). Williams contends that labels such as comedy or drama are more broad than the category of super genre, and therefore fall into a category he calls "film type". Similarly, Williams explains that labels such as animation and musical are more specific to storytelling technique and therefore fall into his category of "voice". For example, according to Williams, a film like Blazing Saddles could be categorized as a comedy (type) Western (super-genre) musical (voice), while Anomalisa is a drama (type) Slice of Life (super-genre) animation (voice). Williams has created a seven-tiered categorization for narrative feature films called the Screenwriters Taxonomy.

A genre movie is a film that follows some or all of the conventions of a particular genre, whether or not it was intentional when the movie was produced.

Film in the context of history
In order to understand the creation and context of each film genre, we must look at its popularity in the context of its place in history. For example, the 1970s Blaxploitation films have been called an attempt to "undermine the rise of Afro-American's Black consciousness movement" of that era. In William Park's analysis of film noir, he states that we must view and interpret film for its message with the context of history within our minds; he states that this is how film can truly be understood by its audience. Film genres such as film noir and Western film reflect values of the time period. While film noir combines German expressionist filming strategies with post World War II ideals; Western films focused on the ideal of the early 20th century. Films such as the musical were created as a form of entertainment during the Great Depression allowing its viewers an escape during tough times. So when watching and analyzing film genres we must remember to remember its true intentions aside from its entertainment value.

Over time, a genre can change through stages: the classic genre era; the parody of the classics; the period where filmmakers deny that their films are part of a certain genre; and finally a critique of the entire genre. This pattern can be seen with the Western film. In the earliest, classic Westerns, there was a clear hero who protected society from lawless villains who lived in the wilderness and came into civilization to commit crimes. However, in revisionist Westerns of the 1970s, the protagonist becomes an antihero who lives in the wilderness to get away from a civilization that is depicted as corrupt, with the villains now integrated into society. Another example of a genre changing over time is the popularity of the neo-noir films in the early 2000s (Mulholland Drive (2001), The Man Who Wasn't There (2001) and Far from Heaven (2002); are these film noir parodies, a repetition of noir genre tropes, or a re-examination of the noir genre?

This is also important to remember when looking at films in the future. As viewers watch a film they are conscious of societal influence with the film itself. In order to understand its true intentions, we must identify its intended audience and what narrative of our current society, as well as it comments to the past in relation with today's society. This enables viewers to understand the evolution of film genres as time and history morphs or views and ideals of the entertainment industry.

title: Film genre