Question: By 19 how many papers had Von Neumann published?

Answer: two major mathematical papers

Although Max insisted von Neumann attend school at the grade level appropriate to his age, he agreed to hire private tutors to give him advanced instruction in those areas in which he had displayed an aptitude. At the age of 15, he began to study advanced calculus under the renowned analyst Gábor Szegő. On their first meeting, Szegő was so astounded with the boy's mathematical talent that he was brought to tears. Some of von Neumann's instant solutions to the problems in calculus posed by Szegő, sketched out on his father's stationery, are still on display at the von Neumann archive in Budapest. <hl> By the age of 19, von Neumann had published two major mathematical papers , the second of which gave the modern definition of ordinal numbers, which superseded Georg Cantor's definition. <hl> At the conclusion of his education at the gymnasium, von Neumann sat for and won the Eötvös Prize, a national prize for mathematics.

Although Max insisted von Neumann attend school at the grade level appropriate to his age, he agreed to hire private tutors to give him advanced instruction in those areas in which he had displayed an aptitude. At the age of 15, he began to study advanced calculus under the renowned analyst Gábor Szegő. On their first meeting, Szegő was so astounded with the boy's mathematical talent that he was brought to tears. Some of von Neumann's instant solutions to the problems in calculus posed by Szegő, sketched out on his father's stationery, are still on display at the von Neumann archive in Budapest. By the age of 19, von Neumann had published <hl> two major mathematical papers <hl>, the second of which gave the modern definition of ordinal numbers, which superseded Georg Cantor's definition. At the conclusion of his education at the gymnasium, von Neumann sat for and won the Eötvös Prize, a national prize for mathematics.

By the age of 19, von Neumann had published <hl> two major mathematical papers <hl> , the second of which gave the modern definition of ordinal numbers, which superseded Georg Cantor's definition.


Question: Von Neumann's modern definition of ordinal numbers superseded whose definition?

Although Max insisted von Neumann attend school at the grade level appropriate to his age, he agreed to hire private tutors to give him advanced instruction in those areas in which he had displayed an aptitude. At the age of 15, he began to study advanced calculus under the renowned analyst Gábor Szegő. On their first meeting, Szegő was so astounded with the boy's mathematical talent that he was brought to tears. Some of von Neumann's instant solutions to the problems in calculus posed by Szegő, sketched out on his father's stationery, are still on display at the von Neumann archive in Budapest. By the age of 19, von Neumann had published two major mathematical papers, the second of which gave the modern definition of ordinal numbers, which superseded Georg Cantor's definition. At the conclusion of his education at the gymnasium, von Neumann sat for and won the Eötvös Prize, a national prize for mathematics.

Answer: Georg Cantor

Although Max insisted von Neumann attend school at the grade level appropriate to his age, he agreed to hire private tutors to give him advanced instruction in those areas in which he had displayed an aptitude. At the age of 15, he began to study advanced calculus under the renowned analyst Gábor Szegő. On their first meeting, Szegő was so astounded with the boy's mathematical talent that he was brought to tears. Some of von Neumann's instant solutions to the problems in calculus posed by Szegő, sketched out on his father's stationery, are still on display at the von Neumann archive in Budapest. <hl> By the age of 19, von Neumann had published two major mathematical papers, the second of which gave the modern definition of ordinal numbers, which superseded Georg Cantor 's <hl> definition. At the conclusion of his education at the gymnasium, von Neumann sat for and won the Eötvös Prize, a national prize for mathematics.

Although Max insisted von Neumann attend school at the grade level appropriate to his age, he agreed to hire private tutors to give him advanced instruction in those areas in which he had displayed an aptitude. At the age of 15, he began to study advanced calculus under the renowned analyst Gábor Szegő. On their first meeting, Szegő was so astounded with the boy's mathematical talent that he was brought to tears. Some of von Neumann's instant solutions to the problems in calculus posed by Szegő, sketched out on his father's stationery, are still on display at the von Neumann archive in Budapest. By the age of 19, von Neumann had published two major mathematical papers, the second of which gave the modern definition of ordinal numbers, which superseded <hl> Georg Cantor <hl>'s definition. At the conclusion of his education at the gymnasium, von Neumann sat for and won the Eötvös Prize, a national prize for mathematics.

By the age of 19, von Neumann had published two major mathematical papers, the second of which gave the modern definition of ordinal numbers, which superseded <hl> Georg Cantor <hl> 's


Question: What prize did Von Neuman win at the conclusion of his gymansium formal education?

Answer: Eötvös Prize, a national prize for mathematics

Although Max insisted von Neumann attend school at the grade level appropriate to his age, he agreed to hire private tutors to give him advanced instruction in those areas in which he had displayed an aptitude. At the age of 15, he began to study advanced calculus under the renowned analyst Gábor Szegő. On their first meeting, Szegő was so astounded with the boy's mathematical talent that he was brought to tears. Some of von Neumann's instant solutions to the problems in calculus posed by Szegő, sketched out on his father's stationery, are still on display at the von Neumann archive in Budapest. By the age of 19, von Neumann had published two major mathematical papers, the second of which gave the modern definition of ordinal numbers, which superseded Georg Cantor's definition. <hl> At the conclusion of his education at the gymnasium, von Neumann sat for and won the Eötvös Prize, a national prize for mathematics . <hl>

Although Max insisted von Neumann attend school at the grade level appropriate to his age, he agreed to hire private tutors to give him advanced instruction in those areas in which he had displayed an aptitude. At the age of 15, he began to study advanced calculus under the renowned analyst Gábor Szegő. On their first meeting, Szegő was so astounded with the boy's mathematical talent that he was brought to tears. Some of von Neumann's instant solutions to the problems in calculus posed by Szegő, sketched out on his father's stationery, are still on display at the von Neumann archive in Budapest. By the age of 19, von Neumann had published two major mathematical papers, the second of which gave the modern definition of ordinal numbers, which superseded Georg Cantor's definition. At the conclusion of his education at the gymnasium, von Neumann sat for and won the <hl> Eötvös Prize, a national prize for mathematics <hl>.

At the conclusion of his education at the gymnasium, von Neumann sat for and won the <hl> Eötvös Prize, a national prize for mathematics <hl> .


Question: What career path was decided upon for Von Neuman?

Since there were few posts in Hungary for mathematicians, and those were not well-paid, his father wanted von Neumann to follow him into industry and therefore invest his time in a more financially useful endeavor than mathematics. So it was decided that the best career path was to become a chemical engineer. This was not something that von Neumann had much knowledge of, so it was arranged for him to take a two-year non-degree course in chemistry at the University of Berlin, after which he sat the entrance exam to the prestigious ETH Zurich, which he passed in September 1923. At the same time, von Neumann also entered Pázmány Péter University in Budapest, as a Ph.D. candidate in mathematics. For his thesis, he chose to produce an axiomatization of Cantor's set theory. He passed his final examinations for his Ph.D. soon after graduating from ETH Zurich in 1926. He then went to the University of Göttingen on a grant from the Rockefeller Foundation to study mathematics under David Hilbert.

Answer: chemical engineer

a chemical engineer .

Since there were few posts in Hungary for mathematicians, and those were not well-paid, his father wanted von Neumann to follow him into industry and therefore invest his time in a more financially useful endeavor than mathematics. So it was decided that the best career path was to become <hl> a chemical engineer . <hl> This was not something that von Neumann had much knowledge of, so it was arranged for him to take a two-year non-degree course in chemistry at the University of Berlin, after which he sat the entrance exam to the prestigious ETH Zurich, which he passed in September 1923. At the same time, von Neumann also entered Pázmány Péter University in Budapest, as a Ph.D. candidate in mathematics. For his thesis, he chose to produce an axiomatization of Cantor's set theory. He passed his final examinations for his Ph.D. soon after graduating from ETH Zurich in 1926. He then went to the University of Göttingen on a grant from the Rockefeller Foundation to study mathematics under David Hilbert.

Since there were few posts in Hungary for mathematicians, and those were not well-paid, his father wanted von Neumann to follow him into industry and therefore invest his time in a more financially useful endeavor than mathematics. So it was decided that the best career path was to become a <hl> chemical engineer <hl>. This was not something that von Neumann had much knowledge of, so it was arranged for him to take a two-year non-degree course in chemistry at the University of Berlin, after which he sat the entrance exam to the prestigious ETH Zurich, which he passed in September 1923. At the same time, von Neumann also entered Pázmány Péter University in Budapest, as a Ph.D. candidate in mathematics. For his thesis, he chose to produce an axiomatization of Cantor's set theory. He passed his final examinations for his Ph.D. soon after graduating from ETH Zurich in 1926. He then went to the University of Göttingen on a grant from the Rockefeller Foundation to study mathematics under David Hilbert.


Question: At what university was it arranged for Von Neumann to enter a chemistry program?

Answer: University of Berlin

Since there were few posts in Hungary for mathematicians, and those were not well-paid, his father wanted von Neumann to follow him into industry and therefore invest his time in a more financially useful endeavor than mathematics. So it was decided that the best career path was to become a chemical engineer. <hl> This was not something that von Neumann had much knowledge of, so it was arranged for him to take a two-year non-degree course in chemistry at the University of Berlin , after which he sat the entrance exam to the prestigious ETH Zurich, which he passed in September 1923. <hl> At the same time, von Neumann also entered Pázmány Péter University in Budapest, as a Ph.D. candidate in mathematics. For his thesis, he chose to produce an axiomatization of Cantor's set theory. He passed his final examinations for his Ph.D. soon after graduating from ETH Zurich in 1926. He then went to the University of Göttingen on a grant from the Rockefeller Foundation to study mathematics under David Hilbert.

Since there were few posts in Hungary for mathematicians, and those were not well-paid, his father wanted von Neumann to follow him into industry and therefore invest his time in a more financially useful endeavor than mathematics. So it was decided that the best career path was to become a chemical engineer. This was not something that von Neumann had much knowledge of, so it was arranged for him to take a two-year non-degree course in chemistry at the <hl> University of Berlin <hl>, after which he sat the entrance exam to the prestigious ETH Zurich, which he passed in September 1923. At the same time, von Neumann also entered Pázmány Péter University in Budapest, as a Ph.D. candidate in mathematics. For his thesis, he chose to produce an axiomatization of Cantor's set theory. He passed his final examinations for his Ph.D. soon after graduating from ETH Zurich in 1926. He then went to the University of Göttingen on a grant from the Rockefeller Foundation to study mathematics under David Hilbert.

This was not something that von Neumann had much knowledge of, so it was arranged for him to take a two-year non-degree course in chemistry at the <hl> University of Berlin <hl> , after which he sat the entrance exam to the prestigious ETH Zurich, which he passed in September 1923.


Question: Where did Von Neumann study as a PHD candidate?

Answer: Pázmány Péter University in Budapest

Since there were few posts in Hungary for mathematicians, and those were not well-paid, his father wanted von Neumann to follow him into industry and therefore invest his time in a more financially useful endeavor than mathematics. So it was decided that the best career path was to become a chemical engineer. This was not something that von Neumann had much knowledge of, so it was arranged for him to take a two-year non-degree course in chemistry at the University of Berlin, after which he sat the entrance exam to the prestigious ETH Zurich, which he passed in September 1923. <hl> At the same time, von Neumann also entered Pázmány Péter University in Budapest , as a Ph.D. candidate in mathematics. <hl> For his thesis, he chose to produce an axiomatization of Cantor's set theory. He passed his final examinations for his Ph.D. soon after graduating from ETH Zurich in 1926. He then went to the University of Göttingen on a grant from the Rockefeller Foundation to study mathematics under David Hilbert.

Since there were few posts in Hungary for mathematicians, and those were not well-paid, his father wanted von Neumann to follow him into industry and therefore invest his time in a more financially useful endeavor than mathematics. So it was decided that the best career path was to become a chemical engineer. This was not something that von Neumann had much knowledge of, so it was arranged for him to take a two-year non-degree course in chemistry at the University of Berlin, after which he sat the entrance exam to the prestigious ETH Zurich, which he passed in September 1923. At the same time, von Neumann also entered <hl> Pázmány Péter University in Budapest <hl>, as a Ph.D. candidate in mathematics. For his thesis, he chose to produce an axiomatization of Cantor's set theory. He passed his final examinations for his Ph.D. soon after graduating from ETH Zurich in 1926. He then went to the University of Göttingen on a grant from the Rockefeller Foundation to study mathematics under David Hilbert.

At the same time, von Neumann also entered <hl> Pázmány Péter University in Budapest <hl> , as a Ph.D. candidate in mathematics.


Question: What was the topic of Von Neuman's thesis?

Answer: axiomatization of Cantor's set theory

Since there were few posts in Hungary for mathematicians, and those were not well-paid, his father wanted von Neumann to follow him into industry and therefore invest his time in a more financially useful endeavor than mathematics. So it was decided that the best career path was to become a chemical engineer. This was not something that von Neumann had much knowledge of, so it was arranged for him to take a two-year non-degree course in chemistry at the University of Berlin, after which he sat the entrance exam to the prestigious ETH Zurich, which he passed in September 1923. At the same time, von Neumann also entered Pázmány Péter University in Budapest, as a Ph.D. candidate in mathematics. <hl> For his thesis, he chose to produce an axiomatization of Cantor's set theory . <hl> He passed his final examinations for his Ph.D. soon after graduating from ETH Zurich in 1926. He then went to the University of Göttingen on a grant from the Rockefeller Foundation to study mathematics under David Hilbert.

Since there were few posts in Hungary for mathematicians, and those were not well-paid, his father wanted von Neumann to follow him into industry and therefore invest his time in a more financially useful endeavor than mathematics. So it was decided that the best career path was to become a chemical engineer. This was not something that von Neumann had much knowledge of, so it was arranged for him to take a two-year non-degree course in chemistry at the University of Berlin, after which he sat the entrance exam to the prestigious ETH Zurich, which he passed in September 1923. At the same time, von Neumann also entered Pázmány Péter University in Budapest, as a Ph.D. candidate in mathematics. For his thesis, he chose to produce an <hl> axiomatization of Cantor's set theory <hl>. He passed his final examinations for his Ph.D. soon after graduating from ETH Zurich in 1926. He then went to the University of Göttingen on a grant from the Rockefeller Foundation to study mathematics under David Hilbert.

For his thesis, he chose to produce an <hl> axiomatization of Cantor's set theory <hl> .


Question: Where did Von Neumann begin to lecture in 1928?

Von Neumann's habilitation was completed on December 13, 1927, and he started his lectures as a privatdozent at the University of Berlin in 1928. By the end of 1927, von Neumann had published twelve major papers in mathematics, and by the end of 1929, thirty-two papers, at a rate of nearly one major paper per month. His reputed powers of speedy, massive memorization and recall allowed him to recite volumes of information, and even entire directories, with ease. In 1929, he briefly became a privatdozent at the University of Hamburg, where the prospects of becoming a tenured professor were better, but in October of that year a better offer presented itself when he was invited to Princeton University in Princeton, New Jersey.

Answer: University of Berlin

<hl> Von Neumann's habilitation was completed on December 13, 1927, and he started his lectures as a privatdozent at the University of Berlin in 1928. <hl> By the end of 1927, von Neumann had published twelve major papers in mathematics, and by the end of 1929, thirty-two papers, at a rate of nearly one major paper per month. His reputed powers of speedy, massive memorization and recall allowed him to recite volumes of information, and even entire directories, with ease. In 1929, he briefly became a privatdozent at the University of Hamburg, where the prospects of becoming a tenured professor were better, but in October of that year a better offer presented itself when he was invited to Princeton University in Princeton, New Jersey.

Von Neumann's habilitation was completed on December 13, 1927, and he started his lectures as a privatdozent at the <hl> University of Berlin <hl> in 1928. By the end of 1927, von Neumann had published twelve major papers in mathematics, and by the end of 1929, thirty-two papers, at a rate of nearly one major paper per month. His reputed powers of speedy, massive memorization and recall allowed him to recite volumes of information, and even entire directories, with ease. In 1929, he briefly became a privatdozent at the University of Hamburg, where the prospects of becoming a tenured professor were better, but in October of that year a better offer presented itself when he was invited to Princeton University in Princeton, New Jersey.


Question: By the end of 1927 how many papers had Von Neuman written?

Answer: twelve major papers in mathematics

Von Neumann's habilitation was completed on December 13, 1927, and he started his lectures as a privatdozent at the University of Berlin in 1928. <hl> By the end of 1927, von Neumann had published twelve major papers in mathematics , and by the end of 1929, thirty-two papers, at a rate of nearly one major paper per month. <hl> His reputed powers of speedy, massive memorization and recall allowed him to recite volumes of information, and even entire directories, with ease. In 1929, he briefly became a privatdozent at the University of Hamburg, where the prospects of becoming a tenured professor were better, but in October of that year a better offer presented itself when he was invited to Princeton University in Princeton, New Jersey.

Von Neumann's habilitation was completed on December 13, 1927, and he started his lectures as a privatdozent at the University of Berlin in 1928. By the end of 1927, von Neumann had published <hl> twelve major papers in mathematics <hl>, and by the end of 1929, thirty-two papers, at a rate of nearly one major paper per month. His reputed powers of speedy, massive memorization and recall allowed him to recite volumes of information, and even entire directories, with ease. In 1929, he briefly became a privatdozent at the University of Hamburg, where the prospects of becoming a tenured professor were better, but in October of that year a better offer presented itself when he was invited to Princeton University in Princeton, New Jersey.

By the end of 1927, von Neumann had published <hl> twelve major papers in mathematics <hl> , and by the end of 1929, thirty-two papers, at a rate of nearly one major paper per month.


Question: How many papers had Von Neumann written by 1929?

Answer: thirty-two papers

Von Neumann's habilitation was completed on December 13, 1927, and he started his lectures as a privatdozent at the University of Berlin in 1928. <hl> By the end of 1927, von Neumann had published twelve major papers in mathematics, and by the end of 1929, thirty-two papers , at a rate of nearly one major paper per month. <hl> His reputed powers of speedy, massive memorization and recall allowed him to recite volumes of information, and even entire directories, with ease. In 1929, he briefly became a privatdozent at the University of Hamburg, where the prospects of becoming a tenured professor were better, but in October of that year a better offer presented itself when he was invited to Princeton University in Princeton, New Jersey.

Von Neumann's habilitation was completed on December 13, 1927, and he started his lectures as a privatdozent at the University of Berlin in 1928. By the end of 1927, von Neumann had published twelve major papers in mathematics, and by the end of 1929, <hl> thirty-two papers <hl>, at a rate of nearly one major paper per month. His reputed powers of speedy, massive memorization and recall allowed him to recite volumes of information, and even entire directories, with ease. In 1929, he briefly became a privatdozent at the University of Hamburg, where the prospects of becoming a tenured professor were better, but in October of that year a better offer presented itself when he was invited to Princeton University in Princeton, New Jersey.

By the end of 1927, von Neumann had published twelve major papers in mathematics, and by the end of 1929, <hl> thirty-two papers <hl> , at a rate of nearly one major paper per month.


Question: What better offer came for Von Neumann in 1930?

Answer: Princeton University

Von Neumann's habilitation was completed on December 13, 1927, and he started his lectures as a privatdozent at the University of Berlin in 1928. By the end of 1927, von Neumann had published twelve major papers in mathematics, and by the end of 1929, thirty-two papers, at a rate of nearly one major paper per month. His reputed powers of speedy, massive memorization and recall allowed him to recite volumes of information, and even entire directories, with ease. <hl> In 1929, he briefly became a privatdozent at the University of Hamburg, where the prospects of becoming a tenured professor were better, but in October of that year a better offer presented itself when he was invited to Princeton University in Princeton, New Jersey. <hl>

Von Neumann's habilitation was completed on December 13, 1927, and he started his lectures as a privatdozent at the University of Berlin in 1928. By the end of 1927, von Neumann had published twelve major papers in mathematics, and by the end of 1929, thirty-two papers, at a rate of nearly one major paper per month. His reputed powers of speedy, massive memorization and recall allowed him to recite volumes of information, and even entire directories, with ease. In 1929, he briefly became a privatdozent at the University of Hamburg, where the prospects of becoming a tenured professor were better, but in October of that year a better offer presented itself when he was invited to <hl> Princeton University <hl> in Princeton, New Jersey.


Question: When did Von Neumann get married?

On New Year's Day in 1930, von Neumann married Mariette Kövesi, who had studied economics at the Budapest University. Before his marriage he was baptized a Catholic. Max had died in 1929. None of the family had converted to Christianity while he was alive, but afterwards they all did. They had one child, a daughter, Marina, who is now a distinguished professor of business administration and public policy at the University of Michigan. The couple divorced in 1937. In October 1938, von Neumann married Klara Dan, whom he had met during his last trips back to Budapest prior to the outbreak of World War II.

Answer: New Year's Day in 1930

<hl> On New Year's Day in 1930 , von Neumann married Mariette Kövesi, who had studied economics at the Budapest University. <hl> Before his marriage he was baptized a Catholic. Max had died in 1929. None of the family had converted to Christianity while he was alive, but afterwards they all did. They had one child, a daughter, Marina, who is now a distinguished professor of business administration and public policy at the University of Michigan. The couple divorced in 1937. In October 1938, von Neumann married Klara Dan, whom he had met during his last trips back to Budapest prior to the outbreak of World War II.

On <hl> New Year's Day in 1930 <hl>, von Neumann married Mariette Kövesi, who had studied economics at the Budapest University. Before his marriage he was baptized a Catholic. Max had died in 1929. None of the family had converted to Christianity while he was alive, but afterwards they all did. They had one child, a daughter, Marina, who is now a distinguished professor of business administration and public policy at the University of Michigan. The couple divorced in 1937. In October 1938, von Neumann married Klara Dan, whom he had met during his last trips back to Budapest prior to the outbreak of World War II.

On <hl> New Year's Day in 1930 <hl> , von Neumann married Mariette Kövesi, who had studied economics at the Budapest University.


Question: What was Von Neumann's wife's name?

Answer: Mariette Kövesi

<hl> On New Year's Day in 1930, von Neumann married Mariette Kövesi , who had studied economics at the Budapest University. <hl> Before his marriage he was baptized a Catholic. Max had died in 1929. None of the family had converted to Christianity while he was alive, but afterwards they all did. They had one child, a daughter, Marina, who is now a distinguished professor of business administration and public policy at the University of Michigan. The couple divorced in 1937. In October 1938, von Neumann married Klara Dan, whom he had met during his last trips back to Budapest prior to the outbreak of World War II.

On New Year's Day in 1930, von Neumann married <hl> Mariette Kövesi <hl>, who had studied economics at the Budapest University. Before his marriage he was baptized a Catholic. Max had died in 1929. None of the family had converted to Christianity while he was alive, but afterwards they all did. They had one child, a daughter, Marina, who is now a distinguished professor of business administration and public policy at the University of Michigan. The couple divorced in 1937. In October 1938, von Neumann married Klara Dan, whom he had met during his last trips back to Budapest prior to the outbreak of World War II.

On New Year's Day in 1930, von Neumann married <hl> Mariette Kövesi <hl> , who had studied economics at the Budapest University.


Question: Before marriage what faith did Von Neumann join?

Answer: he was baptized a Catholic.

On New Year's Day in 1930, von Neumann married Mariette Kövesi, who had studied economics at the Budapest University. <hl> Before his marriage he was baptized a Catholic. Max had died in 1929. <hl> None of the family had converted to Christianity while he was alive, but afterwards they all did. They had one child, a daughter, Marina, who is now a distinguished professor of business administration and public policy at the University of Michigan. The couple divorced in 1937. In October 1938, von Neumann married Klara Dan, whom he had met during his last trips back to Budapest prior to the outbreak of World War II.

On New Year's Day in 1930, von Neumann married Mariette Kövesi, who had studied economics at the Budapest University. Before his marriage <hl> he was baptized a Catholic. <hl> Max had died in 1929. None of the family had converted to Christianity while he was alive, but afterwards they all did. They had one child, a daughter, Marina, who is now a distinguished professor of business administration and public policy at the University of Michigan. The couple divorced in 1937. In October 1938, von Neumann married Klara Dan, whom he had met during his last trips back to Budapest prior to the outbreak of World War II.


Question: What was Von Neumann's child's name?

Answer: Marina

On New Year's Day in 1930, von Neumann married Mariette Kövesi, who had studied economics at the Budapest University. Before his marriage he was baptized a Catholic. Max had died in 1929. None of the family had converted to Christianity while he was alive, but afterwards they all did. <hl> They had one child, a daughter, Marina , who is now a distinguished professor of business administration and public policy at the University of Michigan. <hl> The couple divorced in 1937. In October 1938, von Neumann married Klara Dan, whom he had met during his last trips back to Budapest prior to the outbreak of World War II.

On New Year's Day in 1930, von Neumann married Mariette Kövesi, who had studied economics at the Budapest University. Before his marriage he was baptized a Catholic. Max had died in 1929. None of the family had converted to Christianity while he was alive, but afterwards they all did. They had one child, a daughter, <hl> Marina <hl>, who is now a distinguished professor of business administration and public policy at the University of Michigan. The couple divorced in 1937. In October 1938, von Neumann married Klara Dan, whom he had met during his last trips back to Budapest prior to the outbreak of World War II.

They had one child, a daughter, <hl> Marina <hl> , who is now a distinguished professor of business administration and public policy at the University of Michigan.


Question: When did Von Neumann get married for a second time?

Answer: October 1938

On New Year's Day in 1930, von Neumann married Mariette Kövesi, who had studied economics at the Budapest University. Before his marriage he was baptized a Catholic. Max had died in 1929. None of the family had converted to Christianity while he was alive, but afterwards they all did. They had one child, a daughter, Marina, who is now a distinguished professor of business administration and public policy at the University of Michigan. The couple divorced in 1937. <hl> In October 1938 , von Neumann married Klara Dan, whom he had met during his last trips back to Budapest prior to the outbreak of World War II. <hl>

On New Year's Day in 1930, von Neumann married Mariette Kövesi, who had studied economics at the Budapest University. Before his marriage he was baptized a Catholic. Max had died in 1929. None of the family had converted to Christianity while he was alive, but afterwards they all did. They had one child, a daughter, Marina, who is now a distinguished professor of business administration and public policy at the University of Michigan. The couple divorced in 1937. In <hl> October 1938 <hl>, von Neumann married Klara Dan, whom he had met during his last trips back to Budapest prior to the outbreak of World War II.

In <hl> October 1938 <hl> , von Neumann married Klara Dan, whom he had met during his last trips back to Budapest prior to the outbreak of World War II.


Question: In what year was Von Neumann offered a lifetime professorship?

In 1933, von Neumann was offered a lifetime professorship on the faculty of the Institute for Advanced Study when the institute's plan to appoint Hermann Weyl fell through. He remained a mathematics professor there until his death, although he announced that shortly before his intention to resign and become a professor at large at the University of California. His mother, brothers and in-laws followed John to the United States in 1939. Von Neumann anglicized his first name to John, keeping the German-aristocratic surname of von Neumann. His brothers changed theirs to "Neumann" and "Vonneumann". Von Neumann became a naturalized citizen of the United States in 1937, and immediately tried to become a lieutenant in the United States Army's Officers Reserve Corps. He passed the exams easily, but was ultimately rejected because of his age. His prewar analysis is often quoted. Asked about how France would stand up to Germany he said "Oh, France won't matter."

Answer: 1933

<hl> In 1933 , von Neumann was offered a lifetime professorship on the faculty of the Institute for Advanced Study when the institute's plan to appoint Hermann Weyl fell through. <hl> He remained a mathematics professor there until his death, although he announced that shortly before his intention to resign and become a professor at large at the University of California. His mother, brothers and in-laws followed John to the United States in 1939. Von Neumann anglicized his first name to John, keeping the German-aristocratic surname of von Neumann. His brothers changed theirs to "Neumann" and "Vonneumann". Von Neumann became a naturalized citizen of the United States in 1937, and immediately tried to become a lieutenant in the United States Army's Officers Reserve Corps. He passed the exams easily, but was ultimately rejected because of his age. His prewar analysis is often quoted. Asked about how France would stand up to Germany he said "Oh, France won't matter."

In <hl> 1933 <hl>, von Neumann was offered a lifetime professorship on the faculty of the Institute for Advanced Study when the institute's plan to appoint Hermann Weyl fell through. He remained a mathematics professor there until his death, although he announced that shortly before his intention to resign and become a professor at large at the University of California. His mother, brothers and in-laws followed John to the United States in 1939. Von Neumann anglicized his first name to John, keeping the German-aristocratic surname of von Neumann. His brothers changed theirs to "Neumann" and "Vonneumann". Von Neumann became a naturalized citizen of the United States in 1937, and immediately tried to become a lieutenant in the United States Army's Officers Reserve Corps. He passed the exams easily, but was ultimately rejected because of his age. His prewar analysis is often quoted. Asked about how France would stand up to Germany he said "Oh, France won't matter."

In <hl> 1933 <hl> , von Neumann was offered a lifetime professorship on the faculty of the Institute for Advanced Study when the institute's plan to appoint Hermann Weyl fell through.


Question: In what year did Von Neumann's mother and siblings join him in U.S?

Answer: 1939

In 1933, von Neumann was offered a lifetime professorship on the faculty of the Institute for Advanced Study when the institute's plan to appoint Hermann Weyl fell through. He remained a mathematics professor there until his death, although he announced that shortly before his intention to resign and become a professor at large at the University of California. <hl> His mother, brothers and in-laws followed John to the United States in 1939 . <hl> Von Neumann anglicized his first name to John, keeping the German-aristocratic surname of von Neumann. His brothers changed theirs to "Neumann" and "Vonneumann". Von Neumann became a naturalized citizen of the United States in 1937, and immediately tried to become a lieutenant in the United States Army's Officers Reserve Corps. He passed the exams easily, but was ultimately rejected because of his age. His prewar analysis is often quoted. Asked about how France would stand up to Germany he said "Oh, France won't matter."

In 1933, von Neumann was offered a lifetime professorship on the faculty of the Institute for Advanced Study when the institute's plan to appoint Hermann Weyl fell through. He remained a mathematics professor there until his death, although he announced that shortly before his intention to resign and become a professor at large at the University of California. His mother, brothers and in-laws followed John to the United States in <hl> 1939 <hl>. Von Neumann anglicized his first name to John, keeping the German-aristocratic surname of von Neumann. His brothers changed theirs to "Neumann" and "Vonneumann". Von Neumann became a naturalized citizen of the United States in 1937, and immediately tried to become a lieutenant in the United States Army's Officers Reserve Corps. He passed the exams easily, but was ultimately rejected because of his age. His prewar analysis is often quoted. Asked about how France would stand up to Germany he said "Oh, France won't matter."

His mother, brothers and in-laws followed John to the United States in <hl> 1939 <hl> .


Question: In what year did Von Neumann become a naturalized citizen of US?

Answer: 1937

In 1933, von Neumann was offered a lifetime professorship on the faculty of the Institute for Advanced Study when the institute's plan to appoint Hermann Weyl fell through. He remained a mathematics professor there until his death, although he announced that shortly before his intention to resign and become a professor at large at the University of California. His mother, brothers and in-laws followed John to the United States in 1939. Von Neumann anglicized his first name to John, keeping the German-aristocratic surname of von Neumann. His brothers changed theirs to "Neumann" and "Vonneumann". <hl> Von Neumann became a naturalized citizen of the United States in 1937 , and immediately tried to become a lieutenant in the United States Army's Officers Reserve Corps. <hl> He passed the exams easily, but was ultimately rejected because of his age. His prewar analysis is often quoted. Asked about how France would stand up to Germany he said "Oh, France won't matter."

In 1933, von Neumann was offered a lifetime professorship on the faculty of the Institute for Advanced Study when the institute's plan to appoint Hermann Weyl fell through. He remained a mathematics professor there until his death, although he announced that shortly before his intention to resign and become a professor at large at the University of California. His mother, brothers and in-laws followed John to the United States in 1939. Von Neumann anglicized his first name to John, keeping the German-aristocratic surname of von Neumann. His brothers changed theirs to "Neumann" and "Vonneumann". Von Neumann became a naturalized citizen of the United States in <hl> 1937 <hl>, and immediately tried to become a lieutenant in the United States Army's Officers Reserve Corps. He passed the exams easily, but was ultimately rejected because of his age. His prewar analysis is often quoted. Asked about how France would stand up to Germany he said "Oh, France won't matter."

Von Neumann became a naturalized citizen of the United States in <hl> 1937 <hl> , and immediately tried to become a lieutenant in the United States Army's Officers Reserve Corps.


Question: Why was Neuman rejected from joining US Army?

Answer: age

In 1933, von Neumann was offered a lifetime professorship on the faculty of the Institute for Advanced Study when the institute's plan to appoint Hermann Weyl fell through. He remained a mathematics professor there until his death, although he announced that shortly before his intention to resign and become a professor at large at the University of California. His mother, brothers and in-laws followed John to the United States in 1939. Von Neumann anglicized his first name to John, keeping the German-aristocratic surname of von Neumann. His brothers changed theirs to "Neumann" and "Vonneumann". Von Neumann became a naturalized citizen of the United States in 1937, and immediately tried to become a lieutenant in the United States Army's Officers Reserve Corps. <hl> He passed the exams easily, but was ultimately rejected because of his age . <hl> His prewar analysis is often quoted. Asked about how France would stand up to Germany he said "Oh, France won't matter."

In 1933, von Neumann was offered a lifetime professorship on the faculty of the Institute for Advanced Study when the institute's plan to appoint Hermann Weyl fell through. He remained a mathematics professor there until his death, although he announced that shortly before his intention to resign and become a professor at large at the University of California. His mother, brothers and in-laws followed John to the United States in 1939. Von Neumann anglicized his first name to John, keeping the German-aristocratic surname of von Neumann. His brothers changed theirs to "Neumann" and "Vonneumann". Von Neumann became a naturalized citizen of the United States in 1937, and immediately tried to become a lieutenant in the United States Army's Officers Reserve Corps. He passed the exams easily, but was ultimately rejected because of his <hl> age <hl>. His prewar analysis is often quoted. Asked about how France would stand up to Germany he said "Oh, France won't matter."

He passed the exams easily, but was ultimately rejected because of his <hl> age <hl> .


Question: What type of "off color" humor did Von Neumann especially enjoy?

Von Neumann liked to eat and drink; his wife, Klara, said that he could count everything except calories. He enjoyed Yiddish and "off-color" humor (especially limericks). He was a non-smoker. At Princeton he received complaints for regularly playing extremely loud German march music on his gramophone, which distracted those in neighbouring offices, including Albert Einstein, from their work. Von Neumann did some of his best work blazingly fast in noisy, chaotic environments, and once admonished his wife for preparing a quiet study for him to work in. He never used it, preferring the couple's living room with its television playing loudly.

Answer: limericks

Von Neumann liked to eat and drink; his wife, Klara, said that he could count everything except calories. <hl> He enjoyed Yiddish and "off-color" humor (especially limericks ). <hl> He was a non-smoker. At Princeton he received complaints for regularly playing extremely loud German march music on his gramophone, which distracted those in neighbouring offices, including Albert Einstein, from their work. Von Neumann did some of his best work blazingly fast in noisy, chaotic environments, and once admonished his wife for preparing a quiet study for him to work in. He never used it, preferring the couple's living room with its television playing loudly.

Von Neumann liked to eat and drink; his wife, Klara, said that he could count everything except calories. He enjoyed Yiddish and "off-color" humor (especially <hl> limericks <hl>). He was a non-smoker. At Princeton he received complaints for regularly playing extremely loud German march music on his gramophone, which distracted those in neighbouring offices, including Albert Einstein, from their work. Von Neumann did some of his best work blazingly fast in noisy, chaotic environments, and once admonished his wife for preparing a quiet study for him to work in. He never used it, preferring the couple's living room with its television playing loudly.

He enjoyed Yiddish and "off-color" humor (especially <hl> limericks <hl> ).


Question: What complaints did Von Neumann's neighbor's have?

Answer: regularly playing extremely loud German march music

Von Neumann liked to eat and drink; his wife, Klara, said that he could count everything except calories. He enjoyed Yiddish and "off-color" humor (especially limericks). He was a non-smoker. <hl> At Princeton he received complaints for regularly playing extremely loud German march music on his gramophone, which distracted those in neighbouring offices, including Albert Einstein, from their work. <hl> Von Neumann did some of his best work blazingly fast in noisy, chaotic environments, and once admonished his wife for preparing a quiet study for him to work in. He never used it, preferring the couple's living room with its television playing loudly.

Von Neumann liked to eat and drink; his wife, Klara, said that he could count everything except calories. He enjoyed Yiddish and "off-color" humor (especially limericks). He was a non-smoker. At Princeton he received complaints for <hl> regularly playing extremely loud German march music <hl> on his gramophone, which distracted those in neighbouring offices, including Albert Einstein, from their work. Von Neumann did some of his best work blazingly fast in noisy, chaotic environments, and once admonished his wife for preparing a quiet study for him to work in. He never used it, preferring the couple's living room with its television playing loudly.


Question: What famous neighbor did Von Neumann have that may have been distracted by the music Von Neumann played?

Answer: Albert Einstein,

Von Neumann liked to eat and drink; his wife, Klara, said that he could count everything except calories. He enjoyed Yiddish and "off-color" humor (especially limericks). He was a non-smoker. <hl> At Princeton he received complaints for regularly playing extremely loud German march music on his gramophone, which distracted those in neighbouring offices, including Albert Einstein, from their work. <hl> Von Neumann did some of his best work blazingly fast in noisy, chaotic environments, and once admonished his wife for preparing a quiet study for him to work in. He never used it, preferring the couple's living room with its television playing loudly.

Von Neumann liked to eat and drink; his wife, Klara, said that he could count everything except calories. He enjoyed Yiddish and "off-color" humor (especially limericks). He was a non-smoker. At Princeton he received complaints for regularly playing extremely loud German march music on his gramophone, which distracted those in neighbouring offices, including <hl> Albert Einstein, <hl> from their work. Von Neumann did some of his best work blazingly fast in noisy, chaotic environments, and once admonished his wife for preparing a quiet study for him to work in. He never used it, preferring the couple's living room with its television playing loudly.


Question: What mathematician was Von Nemann's closest friend?

Von Neumann's closest friend in the United States was mathematician Stanislaw Ulam. A later friend of Ulam's, Gian-Carlo Rota writes: "They would spend hours on end gossiping and giggling, swapping Jewish jokes, and drifting in and out of mathematical talk." When von Neumann was dying in hospital, every time Ulam would visit he would come prepared with a new collection of jokes to cheer up his friend. He believed that much of his mathematical thought occurred intuitively, and he would often go to sleep with a problem unsolved, and know the answer immediately upon waking up.

Answer: Stanislaw Ulam

<hl> Von Neumann's closest friend in the United States was mathematician Stanislaw Ulam . <hl> A later friend of Ulam's, Gian-Carlo Rota writes: "They would spend hours on end gossiping and giggling, swapping Jewish jokes, and drifting in and out of mathematical talk." When von Neumann was dying in hospital, every time Ulam would visit he would come prepared with a new collection of jokes to cheer up his friend. He believed that much of his mathematical thought occurred intuitively, and he would often go to sleep with a problem unsolved, and know the answer immediately upon waking up.

Von Neumann's closest friend in the United States was mathematician <hl> Stanislaw Ulam <hl>. A later friend of Ulam's, Gian-Carlo Rota writes: "They would spend hours on end gossiping and giggling, swapping Jewish jokes, and drifting in and out of mathematical talk." When von Neumann was dying in hospital, every time Ulam would visit he would come prepared with a new collection of jokes to cheer up his friend. He believed that much of his mathematical thought occurred intuitively, and he would often go to sleep with a problem unsolved, and know the answer immediately upon waking up.

Von Neumann's closest friend in the United States was mathematician <hl> Stanislaw Ulam <hl> .


Question: What topics of conversation would Ulam and Von Neumann typically share?

Answer: gossiping and giggling, swapping Jewish jokes, and drifting in and out of mathematical talk

Von Neumann's closest friend in the United States was mathematician Stanislaw Ulam. <hl> A later friend of Ulam's, Gian-Carlo Rota writes: "They would spend hours on end gossiping and giggling, swapping Jewish jokes, and drifting in and out of mathematical talk ." <hl> When von Neumann was dying in hospital, every time Ulam would visit he would come prepared with a new collection of jokes to cheer up his friend. He believed that much of his mathematical thought occurred intuitively, and he would often go to sleep with a problem unsolved, and know the answer immediately upon waking up.

Von Neumann's closest friend in the United States was mathematician Stanislaw Ulam. A later friend of Ulam's, Gian-Carlo Rota writes: "They would spend hours on end <hl> gossiping and giggling, swapping Jewish jokes, and drifting in and out of mathematical talk <hl>." When von Neumann was dying in hospital, every time Ulam would visit he would come prepared with a new collection of jokes to cheer up his friend. He believed that much of his mathematical thought occurred intuitively, and he would often go to sleep with a problem unsolved, and know the answer immediately upon waking up.

A later friend of Ulam's, Gian-Carlo Rota writes: "They would spend hours on end <hl> gossiping and giggling, swapping Jewish jokes, and drifting in and out of mathematical talk <hl> ."


Question: What would Ulam use to cheer up Von Neumann in the hospital?

Answer: a new collection of jokes

Von Neumann's closest friend in the United States was mathematician Stanislaw Ulam. A later friend of Ulam's, Gian-Carlo Rota writes: "They would spend hours on end gossiping and giggling, swapping Jewish jokes, and drifting in and out of mathematical talk." <hl> When von Neumann was dying in hospital, every time Ulam would visit he would come prepared with a new collection of jokes to cheer up his friend. <hl> He believed that much of his mathematical thought occurred intuitively, and he would often go to sleep with a problem unsolved, and know the answer immediately upon waking up.

Von Neumann's closest friend in the United States was mathematician Stanislaw Ulam. A later friend of Ulam's, Gian-Carlo Rota writes: "They would spend hours on end gossiping and giggling, swapping Jewish jokes, and drifting in and out of mathematical talk." When von Neumann was dying in hospital, every time Ulam would visit he would come prepared with <hl> a new collection of jokes <hl> to cheer up his friend. He believed that much of his mathematical thought occurred intuitively, and he would often go to sleep with a problem unsolved, and know the answer immediately upon waking up.


Question: What caused a setback in naive set theory at the beginning of 20th century?

The axiomatization of mathematics, on the model of Euclid's Elements, had reached new levels of rigour and breadth at the end of the 19th century, particularly in arithmetic, thanks to the axiom schema of Richard Dedekind and Charles Sanders Peirce, and geometry, thanks to David Hilbert. At the beginning of the 20th century, efforts to base mathematics on naive set theory suffered a setback due to Russell's paradox (on the set of all sets that do not belong to themselves). The problem of an adequate axiomatization of set theory was resolved implicitly about twenty years later by Ernst Zermelo and Abraham Fraenkel. Zermelo–Fraenkel set theory provided a series of principles that allowed for the construction of the sets used in the everyday practice of mathematics. But they did not explicitly exclude the possibility of the existence of a set that belongs to itself. In his doctoral thesis of 1925, von Neumann demonstrated two techniques to exclude such sets—the axiom of foundation and the notion of class.

Answer: Russell's paradox

The axiomatization of mathematics, on the model of Euclid's Elements, had reached new levels of rigour and breadth at the end of the 19th century, particularly in arithmetic, thanks to the axiom schema of Richard Dedekind and Charles Sanders Peirce, and geometry, thanks to David Hilbert. <hl> At the beginning of the 20th century, efforts to base mathematics on naive set theory suffered a setback due to Russell's paradox (on the set of all sets that do not belong to themselves). <hl> The problem of an adequate axiomatization of set theory was resolved implicitly about twenty years later by Ernst Zermelo and Abraham Fraenkel. Zermelo–Fraenkel set theory provided a series of principles that allowed for the construction of the sets used in the everyday practice of mathematics. But they did not explicitly exclude the possibility of the existence of a set that belongs to itself. In his doctoral thesis of 1925, von Neumann demonstrated two techniques to exclude such sets—the axiom of foundation and the notion of class.

The axiomatization of mathematics, on the model of Euclid's Elements, had reached new levels of rigour and breadth at the end of the 19th century, particularly in arithmetic, thanks to the axiom schema of Richard Dedekind and Charles Sanders Peirce, and geometry, thanks to David Hilbert. At the beginning of the 20th century, efforts to base mathematics on naive set theory suffered a setback due to <hl> Russell's paradox <hl> (on the set of all sets that do not belong to themselves). The problem of an adequate axiomatization of set theory was resolved implicitly about twenty years later by Ernst Zermelo and Abraham Fraenkel. Zermelo–Fraenkel set theory provided a series of principles that allowed for the construction of the sets used in the everyday practice of mathematics. But they did not explicitly exclude the possibility of the existence of a set that belongs to itself. In his doctoral thesis of 1925, von Neumann demonstrated two techniques to exclude such sets—the axiom of foundation and the notion of class.


Question: Who resolved the problem of adequate axiomatization of set theory?

Answer: Ernst Zermelo and Abraham Fraenkel

The axiomatization of mathematics, on the model of Euclid's Elements, had reached new levels of rigour and breadth at the end of the 19th century, particularly in arithmetic, thanks to the axiom schema of Richard Dedekind and Charles Sanders Peirce, and geometry, thanks to David Hilbert. At the beginning of the 20th century, efforts to base mathematics on naive set theory suffered a setback due to Russell's paradox (on the set of all sets that do not belong to themselves). <hl> The problem of an adequate axiomatization of set theory was resolved implicitly about twenty years later by Ernst Zermelo and Abraham Fraenkel . <hl> Zermelo–Fraenkel set theory provided a series of principles that allowed for the construction of the sets used in the everyday practice of mathematics. But they did not explicitly exclude the possibility of the existence of a set that belongs to itself. In his doctoral thesis of 1925, von Neumann demonstrated two techniques to exclude such sets—the axiom of foundation and the notion of class.

The axiomatization of mathematics, on the model of Euclid's Elements, had reached new levels of rigour and breadth at the end of the 19th century, particularly in arithmetic, thanks to the axiom schema of Richard Dedekind and Charles Sanders Peirce, and geometry, thanks to David Hilbert. At the beginning of the 20th century, efforts to base mathematics on naive set theory suffered a setback due to Russell's paradox (on the set of all sets that do not belong to themselves). The problem of an adequate axiomatization of set theory was resolved implicitly about twenty years later by <hl> Ernst Zermelo and Abraham Fraenkel <hl>. Zermelo–Fraenkel set theory provided a series of principles that allowed for the construction of the sets used in the everyday practice of mathematics. But they did not explicitly exclude the possibility of the existence of a set that belongs to itself. In his doctoral thesis of 1925, von Neumann demonstrated two techniques to exclude such sets—the axiom of foundation and the notion of class.

The problem of an adequate axiomatization of set theory was resolved implicitly about twenty years later by <hl> Ernst Zermelo and Abraham Fraenkel <hl> .


Question: What 2 techniques did Von Neumann use to exclude sets in his doctoral thesis in 1925?

Answer: the axiom of foundation and the notion of class.

The axiomatization of mathematics, on the model of Euclid's Elements, had reached new levels of rigour and breadth at the end of the 19th century, particularly in arithmetic, thanks to the axiom schema of Richard Dedekind and Charles Sanders Peirce, and geometry, thanks to David Hilbert. At the beginning of the 20th century, efforts to base mathematics on naive set theory suffered a setback due to Russell's paradox (on the set of all sets that do not belong to themselves). The problem of an adequate axiomatization of set theory was resolved implicitly about twenty years later by Ernst Zermelo and Abraham Fraenkel. Zermelo–Fraenkel set theory provided a series of principles that allowed for the construction of the sets used in the everyday practice of mathematics. But they did not explicitly exclude the possibility of the existence of a set that belongs to itself. <hl> In his doctoral thesis of 1925, von Neumann demonstrated two techniques to exclude such sets— the axiom of foundation and the notion of class. <hl>

The axiomatization of mathematics, on the model of Euclid's Elements, had reached new levels of rigour and breadth at the end of the 19th century, particularly in arithmetic, thanks to the axiom schema of Richard Dedekind and Charles Sanders Peirce, and geometry, thanks to David Hilbert. At the beginning of the 20th century, efforts to base mathematics on naive set theory suffered a setback due to Russell's paradox (on the set of all sets that do not belong to themselves). The problem of an adequate axiomatization of set theory was resolved implicitly about twenty years later by Ernst Zermelo and Abraham Fraenkel. Zermelo–Fraenkel set theory provided a series of principles that allowed for the construction of the sets used in the everyday practice of mathematics. But they did not explicitly exclude the possibility of the existence of a set that belongs to itself. In his doctoral thesis of 1925, von Neumann demonstrated two techniques to exclude such sets—<hl> the axiom of foundation and the notion of class. <hl>

In his doctoral thesis of 1925, von Neumann demonstrated two techniques to exclude such sets— <hl> the axiom of foundation and the notion of class. <hl>


Question: What is the method to demonstrate that no contradictions were created by the addition of axiom of foundation?

The axiom of foundation established that every set can be constructed from the bottom up in an ordered succession of steps by way of the principles of Zermelo and Fraenkel, in such a manner that if one set belongs to another then the first must necessarily come before the second in the succession, hence excluding the possibility of a set belonging to itself. To demonstrate that the addition of this new axiom to the others did not produce contradictions, von Neumann introduced a method of demonstration, called the method of inner models, which later became an essential instrument in set theory.

Answer: method of inner models

The axiom of foundation established that every set can be constructed from the bottom up in an ordered succession of steps by way of the principles of Zermelo and Fraenkel, in such a manner that if one set belongs to another then the first must necessarily come before the second in the succession, hence excluding the possibility of a set belonging to itself. <hl> To demonstrate that the addition of this new axiom to the others did not produce contradictions, von Neumann introduced a method of demonstration, called the method of inner models , which later became an essential instrument in set theory. <hl>

The axiom of foundation established that every set can be constructed from the bottom up in an ordered succession of steps by way of the principles of Zermelo and Fraenkel, in such a manner that if one set belongs to another then the first must necessarily come before the second in the succession, hence excluding the possibility of a set belonging to itself. To demonstrate that the addition of this new axiom to the others did not produce contradictions, von Neumann introduced a method of demonstration, called the <hl> method of inner models <hl>, which later became an essential instrument in set theory.

To demonstrate that the addition of this new axiom to the others did not produce contradictions, von Neumann introduced a method of demonstration, called the <hl> method of inner models <hl> , which later became an essential instrument in set theory.


Question: Zermelo and Fraenkel's axiom of foundation established that every set be constructed how?

Answer: from the bottom up in an ordered succession of steps

<hl> The axiom of foundation established that every set can be constructed from the bottom up in an ordered succession of steps by way of the principles of Zermelo and Fraenkel, in such a manner that if one set belongs to another then the first must necessarily come before the second in the succession, hence excluding the possibility of a set belonging to itself. <hl> To demonstrate that the addition of this new axiom to the others did not produce contradictions, von Neumann introduced a method of demonstration, called the method of inner models, which later became an essential instrument in set theory.

The axiom of foundation established that every set can be constructed <hl> from the bottom up in an ordered succession of steps <hl> by way of the principles of Zermelo and Fraenkel, in such a manner that if one set belongs to another then the first must necessarily come before the second in the succession, hence excluding the possibility of a set belonging to itself. To demonstrate that the addition of this new axiom to the others did not produce contradictions, von Neumann introduced a method of demonstration, called the method of inner models, which later became an essential instrument in set theory.


Question: What excluded a set from belonging to itself in set theory?

Answer: if one set belongs to another then the first must necessarily come before the second in the succession

<hl> The axiom of foundation established that every set can be constructed from the bottom up in an ordered succession of steps by way of the principles of Zermelo and Fraenkel, in such a manner that if one set belongs to another then the first must necessarily come before the second in the succession , hence excluding the possibility of a set belonging to itself. <hl> To demonstrate that the addition of this new axiom to the others did not produce contradictions, von Neumann introduced a method of demonstration, called the method of inner models, which later became an essential instrument in set theory.

The axiom of foundation established that every set can be constructed from the bottom up in an ordered succession of steps by way of the principles of Zermelo and Fraenkel, in such a manner that <hl> if one set belongs to another then the first must necessarily come before the second in the succession <hl>, hence excluding the possibility of a set belonging to itself. To demonstrate that the addition of this new axiom to the others did not produce contradictions, von Neumann introduced a method of demonstration, called the method of inner models, which later became an essential instrument in set theory.

The axiom of foundation established that every set can be constructed from the bottom up in an ordered succession of steps by way of the principles of Zermelo and Fraenkel, in such a manner that <hl> if one set belongs to another then the first must necessarily come before the second in the succession <hl> , hence excluding the possibility of a set belonging to itself.


Question: What is a proper class?

The second approach to the problem took as its base the notion of class, and defines a set as a class which belongs to other classes, while a proper class is defined as a class which does not belong to other classes. Under the Zermelo–Fraenkel approach, the axioms impede the construction of a set of all sets which do not belong to themselves. In contrast, under the von Neumann approach, the class of all sets which do not belong to themselves can be constructed, but it is a proper class and not a set.

Answer: a class which does not belong to other classes

<hl> The second approach to the problem took as its base the notion of class, and defines a set as a class which belongs to other classes, while a proper class is defined as a class which does not belong to other classes . <hl> Under the Zermelo–Fraenkel approach, the axioms impede the construction of a set of all sets which do not belong to themselves. In contrast, under the von Neumann approach, the class of all sets which do not belong to themselves can be constructed, but it is a proper class and not a set.

The second approach to the problem took as its base the notion of class, and defines a set as a class which belongs to other classes, while a proper class is defined as <hl> a class which does not belong to other classes <hl>. Under the Zermelo–Fraenkel approach, the axioms impede the construction of a set of all sets which do not belong to themselves. In contrast, under the von Neumann approach, the class of all sets which do not belong to themselves can be constructed, but it is a proper class and not a set.

The second approach to the problem took as its base the notion of class, and defines a set as a class which belongs to other classes, while a proper class is defined as <hl> a class which does not belong to other classes <hl> .


Question: Under what approach does an axiom impede a set of all sets which do not belong to themselves?

Answer: Zermelo–Fraenkel

The second approach to the problem took as its base the notion of class, and defines a set as a class which belongs to other classes, while a proper class is defined as a class which does not belong to other classes. <hl> Under the Zermelo–Fraenkel approach, the axioms impede the construction of a set of all sets which do not belong to themselves. <hl> In contrast, under the von Neumann approach, the class of all sets which do not belong to themselves can be constructed, but it is a proper class and not a set.

The second approach to the problem took as its base the notion of class, and defines a set as a class which belongs to other classes, while a proper class is defined as a class which does not belong to other classes. Under the <hl> Zermelo–Fraenkel <hl> approach, the axioms impede the construction of a set of all sets which do not belong to themselves. In contrast, under the von Neumann approach, the class of all sets which do not belong to themselves can be constructed, but it is a proper class and not a set.


Question: Under Von Neumann's approach can a class of all sets that do not belong to themselves be constructed?

Answer: class of all sets which do not belong to themselves can be constructed, but it is a proper class and not a set.

The second approach to the problem took as its base the notion of class, and defines a set as a class which belongs to other classes, while a proper class is defined as a class which does not belong to other classes. Under the Zermelo–Fraenkel approach, the axioms impede the construction of a set of all sets which do not belong to themselves. <hl> In contrast, under the von Neumann approach, the class of all sets which do not belong to themselves can be constructed, but it is a proper class and not a set. <hl>

The second approach to the problem took as its base the notion of class, and defines a set as a class which belongs to other classes, while a proper class is defined as a class which does not belong to other classes. Under the Zermelo–Fraenkel approach, the axioms impede the construction of a set of all sets which do not belong to themselves. In contrast, under the von Neumann approach, the <hl> class of all sets which do not belong to themselves can be constructed, but it is a proper class and not a set. <hl>


Question: In 1930 who gave a strongly negative answer to Von Neuman's approach to the axiomatic system of theory of sets?

With this contribution of von Neumann, the axiomatic system of the theory of sets became fully satisfactory, and the next question was whether or not it was also definitive, and not subject to improvement. A strongly negative answer arrived in September 1930 at the historic mathematical Congress of Königsberg, in which Kurt Gödel announced his first theorem of incompleteness: the usual axiomatic systems are incomplete, in the sense that they cannot prove every truth which is expressible in their language. This result was sufficiently innovative as to confound the majority of mathematicians of the time.

Answer: Kurt Gödel

With this contribution of von Neumann, the axiomatic system of the theory of sets became fully satisfactory, and the next question was whether or not it was also definitive, and not subject to improvement. <hl> A strongly negative answer arrived in September 1930 at the historic mathematical Congress of Königsberg, in which Kurt Gödel announced his first theorem of incompleteness: the usual axiomatic systems are incomplete, in the sense that they cannot prove every truth which is expressible in their language. <hl> This result was sufficiently innovative as to confound the majority of mathematicians of the time.

With this contribution of von Neumann, the axiomatic system of the theory of sets became fully satisfactory, and the next question was whether or not it was also definitive, and not subject to improvement. A strongly negative answer arrived in September 1930 at the historic mathematical Congress of Königsberg, in which <hl> Kurt Gödel <hl> announced his first theorem of incompleteness: the usual axiomatic systems are incomplete, in the sense that they cannot prove every truth which is expressible in their language. This result was sufficiently innovative as to confound the majority of mathematicians of the time.


Question: What was the central theme of Godel's announcement that the axiomatic system was not complete?

Answer: they cannot prove every truth which is expressible in their language

With this contribution of von Neumann, the axiomatic system of the theory of sets became fully satisfactory, and the next question was whether or not it was also definitive, and not subject to improvement. <hl> A strongly negative answer arrived in September 1930 at the historic mathematical Congress of Königsberg, in which Kurt Gödel announced his first theorem of incompleteness: the usual axiomatic systems are incomplete, in the sense that they cannot prove every truth which is expressible in their language . <hl> This result was sufficiently innovative as to confound the majority of mathematicians of the time.

With this contribution of von Neumann, the axiomatic system of the theory of sets became fully satisfactory, and the next question was whether or not it was also definitive, and not subject to improvement. A strongly negative answer arrived in September 1930 at the historic mathematical Congress of Königsberg, in which Kurt Gödel announced his first theorem of incompleteness: the usual axiomatic systems are incomplete, in the sense that <hl> they cannot prove every truth which is expressible in their language <hl>. This result was sufficiently innovative as to confound the majority of mathematicians of the time.

A strongly negative answer arrived in September 1930 at the historic mathematical Congress of Königsberg, in which Kurt Gödel announced his first theorem of incompleteness: the usual axiomatic systems are incomplete, in the sense that <hl> they cannot prove every truth which is expressible in their language <hl> .


Question: Where was Godel's statement about axiomatic system made in 1930?

Answer: Congress of Königsberg

With this contribution of von Neumann, the axiomatic system of the theory of sets became fully satisfactory, and the next question was whether or not it was also definitive, and not subject to improvement. <hl> A strongly negative answer arrived in September 1930 at the historic mathematical Congress of Königsberg , in which Kurt Gödel announced his first theorem of incompleteness: the usual axiomatic systems are incomplete, in the sense that they cannot prove every truth which is expressible in their language. <hl> This result was sufficiently innovative as to confound the majority of mathematicians of the time.

With this contribution of von Neumann, the axiomatic system of the theory of sets became fully satisfactory, and the next question was whether or not it was also definitive, and not subject to improvement. A strongly negative answer arrived in September 1930 at the historic mathematical <hl> Congress of Königsberg <hl>, in which Kurt Gödel announced his first theorem of incompleteness: the usual axiomatic systems are incomplete, in the sense that they cannot prove every truth which is expressible in their language. This result was sufficiently innovative as to confound the majority of mathematicians of the time.

A strongly negative answer arrived in September 1930 at the historic mathematical <hl> Congress of Königsberg <hl> , in which Kurt Gödel announced his first theorem of incompleteness: the usual axiomatic systems are incomplete, in the sense that they cannot prove every truth which is expressible in their language.


Question: How long did it take von Neumann to figure a response to theory of incompleteness?

But von Neumann, who had participated at the Congress, confirmed his fame as an instantaneous thinker, and in less than a month was able to communicate to Gödel himself an interesting consequence of his theorem: namely that the usual axiomatic systems are unable to demonstrate their own consistency. However, Gödel had already discovered this consequence, now known as his second incompleteness theorem and sent von Neumann a preprint of his article containing both incompleteness theorems. Von Neumann acknowledged Gödel's priority in his next letter. He never thought much of "the American system of claiming personal priority for everything."

Answer: less than a month

<hl> But von Neumann, who had participated at the Congress, confirmed his fame as an instantaneous thinker, and in less than a month was able to communicate to Gödel himself an interesting consequence of his theorem: namely that the usual axiomatic systems are unable to demonstrate their own consistency. <hl> However, Gödel had already discovered this consequence, now known as his second incompleteness theorem and sent von Neumann a preprint of his article containing both incompleteness theorems. Von Neumann acknowledged Gödel's priority in his next letter. He never thought much of "the American system of claiming personal priority for everything."

But von Neumann, who had participated at the Congress, confirmed his fame as an instantaneous thinker, and in <hl> less than a month <hl> was able to communicate to Gödel himself an interesting consequence of his theorem: namely that the usual axiomatic systems are unable to demonstrate their own consistency. However, Gödel had already discovered this consequence, now known as his second incompleteness theorem and sent von Neumann a preprint of his article containing both incompleteness theorems. Von Neumann acknowledged Gödel's priority in his next letter. He never thought much of "the American system of claiming personal priority for everything."


Question: What was the consequence of the theory of incompleteness?

Answer: usual axiomatic systems are unable to demonstrate their own consistency

<hl> But von Neumann, who had participated at the Congress, confirmed his fame as an instantaneous thinker, and in less than a month was able to communicate to Gödel himself an interesting consequence of his theorem: namely that the usual axiomatic systems are unable to demonstrate their own consistency . <hl> However, Gödel had already discovered this consequence, now known as his second incompleteness theorem and sent von Neumann a preprint of his article containing both incompleteness theorems. Von Neumann acknowledged Gödel's priority in his next letter. He never thought much of "the American system of claiming personal priority for everything."

But von Neumann, who had participated at the Congress, confirmed his fame as an instantaneous thinker, and in less than a month was able to communicate to Gödel himself an interesting consequence of his theorem: namely that the <hl> usual axiomatic systems are unable to demonstrate their own consistency <hl>. However, Gödel had already discovered this consequence, now known as his second incompleteness theorem and sent von Neumann a preprint of his article containing both incompleteness theorems. Von Neumann acknowledged Gödel's priority in his next letter. He never thought much of "the American system of claiming personal priority for everything."

But von Neumann, who had participated at the Congress, confirmed his fame as an instantaneous thinker, and in less than a month was able to communicate to Gödel himself an interesting consequence of his theorem: namely that the <hl> usual axiomatic systems are unable to demonstrate their own consistency <hl> .


Question: Upon revising the theory of incompleteness, what was the name of the new Godel theory?

Answer: second incompleteness theorem

But von Neumann, who had participated at the Congress, confirmed his fame as an instantaneous thinker, and in less than a month was able to communicate to Gödel himself an interesting consequence of his theorem: namely that the usual axiomatic systems are unable to demonstrate their own consistency. <hl> However, Gödel had already discovered this consequence, now known as his second incompleteness theorem and sent von Neumann a preprint of his article containing both incompleteness theorems. <hl> Von Neumann acknowledged Gödel's priority in his next letter. He never thought much of "the American system of claiming personal priority for everything."

But von Neumann, who had participated at the Congress, confirmed his fame as an instantaneous thinker, and in less than a month was able to communicate to Gödel himself an interesting consequence of his theorem: namely that the usual axiomatic systems are unable to demonstrate their own consistency. However, Gödel had already discovered this consequence, now known as his <hl> second incompleteness theorem <hl> and sent von Neumann a preprint of his article containing both incompleteness theorems. Von Neumann acknowledged Gödel's priority in his next letter. He never thought much of "the American system of claiming personal priority for everything."


Question: What field did Von Neuman establish?

Von Neumann founded the field of continuous geometry. It followed his path-breaking work on rings of operators. In mathematics, continuous geometry is an analogue of complex projective geometry, where instead of the dimension of a subspace being in a discrete set 0, 1, ..., n, it can be an element of the unit interval [0,1]. Von Neumann was motivated by his discovery of von Neumann algebras with a dimension function taking a continuous range of dimensions, and the first example of a continuous geometry other than projective space was the projections of the hyperfinite type II factor.

Answer: continuous geometry.

<hl> Von Neumann founded the field of continuous geometry. It followed his path-breaking work on rings of operators. <hl> In mathematics, continuous geometry is an analogue of complex projective geometry, where instead of the dimension of a subspace being in a discrete set 0, 1, ..., n, it can be an element of the unit interval [0,1]. Von Neumann was motivated by his discovery of von Neumann algebras with a dimension function taking a continuous range of dimensions, and the first example of a continuous geometry other than projective space was the projections of the hyperfinite type II factor.

Von Neumann founded the field of <hl> continuous geometry. <hl> It followed his path-breaking work on rings of operators. In mathematics, continuous geometry is an analogue of complex projective geometry, where instead of the dimension of a subspace being in a discrete set 0, 1, ..., n, it can be an element of the unit interval [0,1]. Von Neumann was motivated by his discovery of von Neumann algebras with a dimension function taking a continuous range of dimensions, and the first example of a continuous geometry other than projective space was the projections of the hyperfinite type II factor.


Question: What is the distinction of continuous geometry?

Answer: instead of the dimension of a subspace being in a discrete set 0, 1, ..., n, it can be an element of the unit interval

Von Neumann founded the field of continuous geometry. It followed his path-breaking work on rings of operators. <hl> In mathematics, continuous geometry is an analogue of complex projective geometry, where instead of the dimension of a subspace being in a discrete set 0, 1, ..., n, it can be an element of the unit interval [0,1]. <hl> Von Neumann was motivated by his discovery of von Neumann algebras with a dimension function taking a continuous range of dimensions, and the first example of a continuous geometry other than projective space was the projections of the hyperfinite type II factor.

Von Neumann founded the field of continuous geometry. It followed his path-breaking work on rings of operators. In mathematics, continuous geometry is an analogue of complex projective geometry, where <hl> instead of the dimension of a subspace being in a discrete set 0, 1, ..., n, it can be an element of the unit interval <hl> [0,1]. Von Neumann was motivated by his discovery of von Neumann algebras with a dimension function taking a continuous range of dimensions, and the first example of a continuous geometry other than projective space was the projections of the hyperfinite type II factor.


Question: What was the first example of continuous geometry?

Answer: projections of the hyperfinite type II factor

Von Neumann founded the field of continuous geometry. It followed his path-breaking work on rings of operators. In mathematics, continuous geometry is an analogue of complex projective geometry, where instead of the dimension of a subspace being in a discrete set 0, 1, ..., n, it can be an element of the unit interval [0,1]. Von Neumann was motivated by his discovery of von Neumann algebras with a dimension function taking a continuous range of dimensions, and the first example of a continuous geometry other than projective space was <hl> the projections of the hyperfinite type II factor . <hl>

Von Neumann founded the field of continuous geometry. It followed his path-breaking work on rings of operators. In mathematics, continuous geometry is an analogue of complex projective geometry, where instead of the dimension of a subspace being in a discrete set 0, 1, ..., n, it can be an element of the unit interval [0,1]. Von Neumann was motivated by his discovery of von Neumann algebras with a dimension function taking a continuous range of dimensions, and the first example of a continuous geometry other than projective space was the <hl> projections of the hyperfinite type II factor <hl>.

the <hl> projections of the hyperfinite type II factor <hl> .


Question: In what way did von Neumann make spectacular contributions to measure theory?

In a series of famous papers, von Neumann made spectacular contributions to measure theory. The work of Banach had implied that the problem of measure has a positive solution if n = 1 or n = 2 and a negative solution in all other cases. Von Neumann's work argued that the "problem is essentially group-theoretic in character, and that, in particular, for the solvability of the problem of measure the ordinary algebraic concept of solvability of a group is relevant. Thus, according to von Neumann, it is the change of group that makes a difference, not the change of space."

Answer: In a series of famous papers

<hl> In a series of famous papers , von Neumann made spectacular contributions to measure theory. <hl> The work of Banach had implied that the problem of measure has a positive solution if n = 1 or n = 2 and a negative solution in all other cases. Von Neumann's work argued that the "problem is essentially group-theoretic in character, and that, in particular, for the solvability of the problem of measure the ordinary algebraic concept of solvability of a group is relevant. Thus, according to von Neumann, it is the change of group that makes a difference, not the change of space."

<hl> In a series of famous papers <hl>, von Neumann made spectacular contributions to measure theory. The work of Banach had implied that the problem of measure has a positive solution if n = 1 or n = 2 and a negative solution in all other cases. Von Neumann's work argued that the "problem is essentially group-theoretic in character, and that, in particular, for the solvability of the problem of measure the ordinary algebraic concept of solvability of a group is relevant. Thus, according to von Neumann, it is the change of group that makes a difference, not the change of space."

<hl> In a series of famous papers <hl> , von Neumann made spectacular contributions to measure theory.


Question: What concept was relevant to the solvability of the problem of measure?

Answer: algebraic concept of solvability of a group is relevant

In a series of famous papers, von Neumann made spectacular contributions to measure theory. The work of Banach had implied that the problem of measure has a positive solution if n = 1 or n = 2 and a negative solution in all other cases. <hl> Von Neumann's work argued that the "problem is essentially group-theoretic in character, and that, in particular, for the solvability of the problem of measure the ordinary algebraic concept of solvability of a group is relevant . <hl> Thus, according to von Neumann, it is the change of group that makes a difference, not the change of space."

In a series of famous papers, von Neumann made spectacular contributions to measure theory. The work of Banach had implied that the problem of measure has a positive solution if n = 1 or n = 2 and a negative solution in all other cases. Von Neumann's work argued that the "problem is essentially group-theoretic in character, and that, in particular, for the solvability of the problem of measure the ordinary <hl> algebraic concept of solvability of a group is relevant <hl>. Thus, according to von Neumann, it is the change of group that makes a difference, not the change of space."

Von Neumann's work argued that the "problem is essentially group-theoretic in character, and that, in particular, for the solvability of the problem of measure the ordinary <hl> algebraic concept of solvability of a group is relevant <hl> .


Question: According to von Neumann what is the most important part of the problem of measure?

Answer: change of group that makes a difference, not the change of space

In a series of famous papers, von Neumann made spectacular contributions to measure theory. The work of Banach had implied that the problem of measure has a positive solution if n = 1 or n = 2 and a negative solution in all other cases. Von Neumann's work argued that the "problem is essentially group-theoretic in character, and that, in particular, for the solvability of the problem of measure the ordinary algebraic concept of solvability of a group is relevant. <hl> Thus, according to von Neumann, it is the change of group that makes a difference, not the change of space ." <hl>

In a series of famous papers, von Neumann made spectacular contributions to measure theory. The work of Banach had implied that the problem of measure has a positive solution if n = 1 or n = 2 and a negative solution in all other cases. Von Neumann's work argued that the "problem is essentially group-theoretic in character, and that, in particular, for the solvability of the problem of measure the ordinary algebraic concept of solvability of a group is relevant. Thus, according to von Neumann, it is the <hl> change of group that makes a difference, not the change of space <hl>."

Thus, according to von Neumann, it is the <hl> change of group that makes a difference, not the change of space <hl> ."


Question: What is often considered more important than the results of von Neumann's papers?

In a number of von Neumann's papers, the methods of argument he employed are considered even more significant than the results. In anticipation of his later study of dimension theory in algebras of operators, von Neumann used results on equivalence by finite decomposition, and reformulated the problem of measure in terms of functions. In his 1936 paper on analytic measure theory, he used the Haar theorem in the solution of Hilbert's fifth problem in the case of compact groups. In 1938, he was awarded the Bôcher Memorial Prize for his work in analysis.

Answer: methods of argument he employed

<hl> In a number of von Neumann's papers, the methods of argument he employed are considered even more significant than the results. <hl> In anticipation of his later study of dimension theory in algebras of operators, von Neumann used results on equivalence by finite decomposition, and reformulated the problem of measure in terms of functions. In his 1936 paper on analytic measure theory, he used the Haar theorem in the solution of Hilbert's fifth problem in the case of compact groups. In 1938, he was awarded the Bôcher Memorial Prize for his work in analysis.

In a number of von Neumann's papers, the <hl> methods of argument he employed <hl> are considered even more significant than the results. In anticipation of his later study of dimension theory in algebras of operators, von Neumann used results on equivalence by finite decomposition, and reformulated the problem of measure in terms of functions. In his 1936 paper on analytic measure theory, he used the Haar theorem in the solution of Hilbert's fifth problem in the case of compact groups. In 1938, he was awarded the Bôcher Memorial Prize for his work in analysis.


Question: In 1938 what was von Neumann awarded?

Answer: Bôcher Memorial Prize

In a number of von Neumann's papers, the methods of argument he employed are considered even more significant than the results. In anticipation of his later study of dimension theory in algebras of operators, von Neumann used results on equivalence by finite decomposition, and reformulated the problem of measure in terms of functions. In his 1936 paper on analytic measure theory, he used the Haar theorem in the solution of Hilbert's fifth problem in the case of compact groups. <hl> In 1938, he was awarded the Bôcher Memorial Prize for his work in analysis. <hl>

In a number of von Neumann's papers, the methods of argument he employed are considered even more significant than the results. In anticipation of his later study of dimension theory in algebras of operators, von Neumann used results on equivalence by finite decomposition, and reformulated the problem of measure in terms of functions. In his 1936 paper on analytic measure theory, he used the Haar theorem in the solution of Hilbert's fifth problem in the case of compact groups. In 1938, he was awarded the <hl> Bôcher Memorial Prize <hl> for his work in analysis.


Question: What did von Neumann use the Haar theorem to solve in a 1936 paper?

Answer: Hilbert's fifth problem in the case of compact groups

In a number of von Neumann's papers, the methods of argument he employed are considered even more significant than the results. In anticipation of his later study of dimension theory in algebras of operators, von Neumann used results on equivalence by finite decomposition, and reformulated the problem of measure in terms of functions. <hl> In his 1936 paper on analytic measure theory, he used the Haar theorem in the solution of Hilbert's fifth problem in the case of compact groups . <hl> In 1938, he was awarded the Bôcher Memorial Prize for his work in analysis.

In a number of von Neumann's papers, the methods of argument he employed are considered even more significant than the results. In anticipation of his later study of dimension theory in algebras of operators, von Neumann used results on equivalence by finite decomposition, and reformulated the problem of measure in terms of functions. In his 1936 paper on analytic measure theory, he used the Haar theorem in the solution of <hl> Hilbert's fifth problem in the case of compact groups <hl>. In 1938, he was awarded the Bôcher Memorial Prize for his work in analysis.

In his 1936 paper on analytic measure theory, he used the Haar theorem in the solution of <hl> Hilbert's fifth problem in the case of compact groups <hl> .


Question: What is von Neumann algebra?

Von Neumann introduced the study of rings of operators, through the von Neumann algebras. A von Neumann algebra is a *-algebra of bounded operators on a Hilbert space that is closed in the weak operator topology and contains the identity operator. The von Neumann bicommutant theorem shows that the analytic definition is equivalent to a purely algebraic definition as an algebra of symmetries. The direct integral was introduced in 1949 by John von Neumann. One of von Neumann's analyses was to reduce the classification of von Neumann algebras on separable Hilbert spaces to the classification of factors.

Answer: algebra of bounded operators on a Hilbert space that is closed in the weak operator topology and contains the identity operator

Von Neumann introduced the study of rings of operators, through the von Neumann algebras. <hl> A von Neumann algebra is a *- algebra of bounded operators on a Hilbert space that is closed in the weak operator topology and contains the identity operator . <hl> The von Neumann bicommutant theorem shows that the analytic definition is equivalent to a purely algebraic definition as an algebra of symmetries. The direct integral was introduced in 1949 by John von Neumann. One of von Neumann's analyses was to reduce the classification of von Neumann algebras on separable Hilbert spaces to the classification of factors.

Von Neumann introduced the study of rings of operators, through the von Neumann algebras. A von Neumann algebra is a *-<hl> algebra of bounded operators on a Hilbert space that is closed in the weak operator topology and contains the identity operator <hl>. The von Neumann bicommutant theorem shows that the analytic definition is equivalent to a purely algebraic definition as an algebra of symmetries. The direct integral was introduced in 1949 by John von Neumann. One of von Neumann's analyses was to reduce the classification of von Neumann algebras on separable Hilbert spaces to the classification of factors.

A von Neumann algebra is a *- <hl> algebra of bounded operators on a Hilbert space that is closed in the weak operator topology and contains the identity operator <hl> .


Question: What does the von Neumann bicommutant theorem show?

Answer: the analytic definition is equivalent to a purely algebraic definition as an algebra of symmetries.

Von Neumann introduced the study of rings of operators, through the von Neumann algebras. A von Neumann algebra is a *-algebra of bounded operators on a Hilbert space that is closed in the weak operator topology and contains the identity operator. <hl> The von Neumann bicommutant theorem shows that the analytic definition is equivalent to a purely algebraic definition as an algebra of symmetries. The direct integral was introduced in 1949 by John von Neumann. <hl> One of von Neumann's analyses was to reduce the classification of von Neumann algebras on separable Hilbert spaces to the classification of factors.

Von Neumann introduced the study of rings of operators, through the von Neumann algebras. A von Neumann algebra is a *-algebra of bounded operators on a Hilbert space that is closed in the weak operator topology and contains the identity operator. The von Neumann bicommutant theorem shows that <hl> the analytic definition is equivalent to a purely algebraic definition as an algebra of symmetries. <hl> The direct integral was introduced in 1949 by John von Neumann. One of von Neumann's analyses was to reduce the classification of von Neumann algebras on separable Hilbert spaces to the classification of factors.


Question: When was direct integral introduced by von Neumann?

Answer: 1949

Von Neumann introduced the study of rings of operators, through the von Neumann algebras. A von Neumann algebra is a *-algebra of bounded operators on a Hilbert space that is closed in the weak operator topology and contains the identity operator. The von Neumann bicommutant theorem shows that the analytic definition is equivalent to a purely algebraic definition as an algebra of symmetries. <hl> The direct integral was introduced in 1949 by John von Neumann. <hl> One of von Neumann's analyses was to reduce the classification of von Neumann algebras on separable Hilbert spaces to the classification of factors.

Von Neumann introduced the study of rings of operators, through the von Neumann algebras. A von Neumann algebra is a *-algebra of bounded operators on a Hilbert space that is closed in the weak operator topology and contains the identity operator. The von Neumann bicommutant theorem shows that the analytic definition is equivalent to a purely algebraic definition as an algebra of symmetries. The direct integral was introduced in <hl> 1949 <hl> by John von Neumann. One of von Neumann's analyses was to reduce the classification of von Neumann algebras on separable Hilbert spaces to the classification of factors.