romikgosai/squad_modified_2 huggingface dataset:
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.

Q1: Where did Von Neumann begin to lecture in 1928?
A1: University of Berlin
Q2: By the end of 1927 how many papers had Von Neuman written?
A2: twelve major papers in mathematics
Q3: How many papers had Von Neumann written by 1929?
A3: thirty-two papers
Q4: What better offer came for Von Neumann in 1930?
A4: Princeton University
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.

Q1: When did Von Neumann get married?
A1: New Year's Day in 1930
Q2: What was Von Neumann's wife's name?
A2: Mariette Kövesi
Q3: Before marriage what faith did Von Neumann join?
A3: he was baptized a Catholic.
Q4: What was Von Neumann's child's name?
A4: Marina
Q5: When did Von Neumann get married for a second time?
A5: October 1938
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."

Q1: In what year was Von Neumann offered a lifetime professorship?
A1: 1933
Q2: In what year did Von Neumann's mother and siblings join him in U.S?
A2: 1939
Q3: In what year did Von Neumann become a naturalized citizen of US?
A3: 1937
Q4: Why was Neuman rejected from joining US Army?
A4: age
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.

Q1: What type of "off color" humor did Von Neumann especially enjoy?
A1: limericks
Q2: What complaints did Von Neumann's neighbor's have?
A2: regularly playing extremely loud German march music
Q3: What famous neighbor did Von Neumann have that may have been distracted by the music Von Neumann played?
A3: Albert Einstein,
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.

Q1: What mathematician was Von Nemann's closest friend?
A1: Stanislaw Ulam
Q2: What topics of conversation would Ulam and Von Neumann typically share?
A2: gossiping and giggling, swapping Jewish jokes, and drifting in and out of mathematical talk
Q3: What would Ulam use to cheer up Von Neumann in the hospital?
A3: a new collection of jokes
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.

Q1: What caused a setback in naive set theory at the beginning of 20th century?
A1: Russell's paradox
Q2: Who resolved the problem of adequate axiomatization of set theory?
A2: Ernst Zermelo and Abraham Fraenkel
Q3: What 2 techniques did Von Neumann use to exclude sets in his doctoral thesis in 1925?
A3: the axiom of foundation and the notion of class.
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.

Q1: What is the method to demonstrate that no contradictions were created by the addition of axiom of foundation?
A1: method of inner models
Q2: Zermelo and Fraenkel's axiom of foundation established that every set be constructed how?
A2: from the bottom up in an ordered succession of steps
Q3: What excluded a set from belonging to itself in set theory?
A3: if one set belongs to another then the first must necessarily come before the second in the succession
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.

Q1: What is a proper class?
A1: a class which does not belong to other classes
Q2: Under what approach does an axiom impede a set of all sets which do not belong to themselves?
A2: Zermelo–Fraenkel
Q3: Under Von Neumann's approach can a class of all sets that do not belong to themselves be constructed?
A3: class of all sets which do not belong to themselves can be constructed, but it is a proper class and not a set.
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.

Q1: In 1930 who gave a strongly negative answer to Von Neuman's approach to the axiomatic system of theory of sets?
A1: Kurt Gödel
Q2: What was the central theme of Godel's announcement that the axiomatic system was not complete?
A2: they cannot prove every truth which is expressible in their language
Q3: Where was Godel's statement about axiomatic system made in 1930?
A3: Congress of Königsberg
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."

Q1: How long did it take von Neumann to figure a response to theory of incompleteness?
A1: less than a month
Q2: What was the consequence of the theory of incompleteness?
A2: usual axiomatic systems are unable to demonstrate their own consistency
Q3: Upon revising the theory of incompleteness, what was the name of the new Godel theory?
A3: second incompleteness theorem
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.

Q1: What field did Von Neuman establish?
A1: continuous geometry.
Q2: What is the distinction of continuous geometry?
A2: instead of the dimension of a subspace being in a discrete set 0, 1, ..., n, it can be an element of the unit interval
Q3: What was the first example of continuous geometry?
A3: projections of the hyperfinite type II factor
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."

Q1: In what way did von Neumann make spectacular contributions to measure theory?
A1: In a series of famous papers
Q2: What concept was relevant to the solvability of the problem of measure?
A2: algebraic concept of solvability of a group is relevant
Q3: According to von Neumann what is the most important part of the problem of measure?
A3: change of group that makes a difference, not the change of space
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.

Q1: What is often considered more important than the results of von Neumann's papers?
A1: methods of argument he employed
Q2: In 1938 what was von Neumann awarded?
A2: Bôcher Memorial Prize
Q3: What did von Neumann use the Haar theorem to solve in a 1936 paper?
A3: Hilbert's fifth problem in the case of compact groups
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.

Q1: What is von Neumann algebra?
A1: algebra of bounded operators on a Hilbert space that is closed in the weak operator topology and contains the identity operator
Q2: What does the von Neumann bicommutant theorem show?
A2: the analytic definition is equivalent to a purely algebraic definition as an algebra of symmetries.
Q3: When was direct integral introduced by von Neumann?
A3: 1949
Q4: What was introduced in von Neumann algebras?
A4: study of rings of operators
Von Neumann worked on lattice theory between 1937 and 1939. Von Neumann provided an abstract exploration of dimension in completed complemented modular topological lattices: "Dimension is determined, up to a positive linear transformation, by the following two properties. It is conserved by perspective mappings ("perspectivities") and ordered by inclusion. The deepest part of the proof concerns the equivalence of perspectivity with "projectivity by decomposition"—of which a corollary is the transitivity of perspectivity." Garrett Birkhoff writes: "John von Neumann's brilliant mind blazed over lattice theory like a meteor".

Q1: What did von Neumann work on in 1937 - 39?
A1: lattice theory
Q2: What properties determine dimension in lattice theory?
A2: conserved by perspective mappings ("perspectivities") and ordered by inclusion
Q3: Who wrote that "John von Neumann's brilliant mind blazed over lattice theory like a meteor?"
A3: Garrett Birkhoff
Additionally, "[I]n the general case, von Neumann proved the following basic representation theorem. Any complemented modular lattice L having a "basis" of n≥4 pairwise perspective elements, is isomorphic with the lattice ℛ(R) of all principal right-ideals of a suitable regular ring R. This conclusion is the culmination of 140 pages of brilliant and incisive algebra involving entirely novel axioms. Anyone wishing to get an unforgettable impression of the razor edge of von Neumann's mind, need merely try to pursue this chain of exact reasoning for himself—realizing that often five pages of it were written down before breakfast, seated at a living room writing-table in a bathrobe."

Q1: What type of axioms were used by von Neumann in his lattice work?
A1: novel axioms
Q2: How long was von Neumann's lattice theory paper?
A2: 140 pages
Q3: In what room did von Neumann often write at home?
A3: living room writing-table
Von Neumann was the first to establish a rigorous mathematical framework for quantum mechanics, known as the Dirac–von Neumann axioms, with his 1932 work Mathematical Foundations of Quantum Mechanics. After having completed the axiomatization of set theory, he began to confront the axiomatization of quantum mechanics. He realized, in 1926, that a state of a quantum system could be represented by a point in a (complex) Hilbert space that, in general, could be infinite-dimensional even for a single particle. In this formalism of quantum mechanics, observable quantities such as position or momentum are represented as linear operators acting on the Hilbert space associated with the quantum system.

Q1: Who established the framework for quantum mechanics?
A1: Von Neumann
Q2: What paper did von Neumann produce in 1932?
A2: Mathematical Foundations of Quantum Mechanics
Q3: What von Neumann work preceded the axiomatiztion of quatum mechanics?
A3: axiomatization of set theory
The physics of quantum mechanics was thereby reduced to the mathematics of Hilbert spaces and linear operators acting on them. For example, the uncertainty principle, according to which the determination of the position of a particle prevents the determination of its momentum and vice versa, is translated into the non-commutativity of the two corresponding operators. This new mathematical formulation included as special cases the formulations of both Heisenberg and Schrödinger. When Heisenberg was informed von Neumann had clarified the difference between an unbounded operator that was a Self-adjoint operator and one that was merely symmetric, Heisenberg replied "Eh? What is the difference?"

Q1: What are the physics of quantum mechanics?
A1: mathematics of Hilbert spaces and linear operators acting on them
Q2: What is the uncertainty principle?
A2: determination of the position of a particle prevents the determination of its momentum and vice versa
Q3: The physics of quantum mechanics included special cases for what work?
A3: formulations of both Heisenberg and Schrödinger
Von Neumann's abstract treatment permitted him also to confront the foundational issue of determinism versus non-determinism, and in the book he presented a proof that the statistical results of quantum mechanics could not possibly be averages of an underlying set of determined "hidden variables," as in classical statistical mechanics. In 1966, John S. Bell published a paper arguing that the proof contained a conceptual error and was therefore invalid. However, in 2010, Jeffrey Bub argued that Bell had misconstrued von Neumann's proof, and pointed out that the proof, though not valid for all hidden variable theories, does rule out a well-defined and important subset. Bub also suggests that von Neumann was aware of this limitation, and that von Neumann did not claim that his proof completely ruled out hidden variable theories.

Q1: Von Neumann's abstract treatment allowed him to work on what foundational issue?
A1: determinism versus non-determinism
Q2: What was John Bell's argument?
A2: proof contained a conceptual error and was therefore invalid
Q3: What did Jeffrey Bub argue in 2010?
A3: Bell had misconstrued von Neumann's proof
Q4: Did von Neumann rule hidden variable theories?
A4: von Neumann did not claim that his proof completely ruled out hidden variable theories
In a chapter of The Mathematical Foundations of Quantum Mechanics, von Neumann deeply analyzed the so-called measurement problem. He concluded that the entire physical universe could be made subject to the universal wave function. Since something "outside the calculation" was needed to collapse the wave function, von Neumann concluded that the collapse was caused by the consciousness of the experimenter (although this view was accepted by Eugene Wigner, the Von Neumann–Wigner interpretation never gained acceptance amongst the majority of physicists).

Q1: What results came from von Neumann's deep analysis of the measurement problem?
A1: physical universe could be made subject to the universal wave function
Q2: What "outside of calculation" variable could initiate collapse in von Neumann's conclusion?
A2: consciousness of the experimenter
Q3: Were von Neumann's conclusions of collapse of entire universe cause accepted widely?
A3: Von Neumann–Wigner interpretation never gained acceptance amongst the majority of physicists
In a famous paper of 1936 with Garrett Birkhoff, the first work ever to introduce quantum logics, von Neumann and Birkhoff first proved that quantum mechanics requires a propositional calculus substantially different from all classical logics and rigorously isolated a new algebraic structure for quantum logics. The concept of creating a propositional calculus for quantum logic was first outlined in a short section in von Neumann's 1932 work, but in 1936, the need for the new propositional calculus was demonstrated through several proofs. For example, photons cannot pass through two successive filters that are polarized perpendicularly (e.g., one horizontally and the other vertically), and therefore, a fortiori, it cannot pass if a third filter polarized diagonally is added to the other two, either before or after them in the succession, but if the third filter is added in between the other two, the photons will, indeed, pass through. This experimental fact is translatable into logic as the non-commutativity of conjunction . It was also demonstrated that the laws of distribution of classical logic,  and , are not valid for quantum theory.

Q1: With whom did von Neuman work on a paper in 1936 that introduce quantum logic?
A1: Garrett Birkhoff
Q2: What concept was created for quatum logic?
A2: propositional calculus
Q3: What is the difference of logic in quantum theory?
A3: laws of distribution of classical logic,  and , are not valid for quantum theory
Von Neumann founded the field of game theory as a mathematical discipline. Von Neumann proved his minimax theorem in 1928. This theorem establishes that in zero-sum games with perfect information (i.e. in which players know at each time all moves that have taken place so far), there exists a pair of strategies for both players that allows each to minimize his maximum losses, hence the name minimax. When examining every possible strategy, a player must consider all the possible responses of his adversary. The player then plays out the strategy that will result in the minimization of his maximum loss.

Q1: What year was game theory established?
A1: 1928
Q2: What discipline is game theory derived from?
A2: mathematical discipline.
Q3: what are the possible strategies in minimax theory?
A3: a pair of strategies for both players that allows each to minimize his maximum losses
Q4: What must a player consider when determining every possible strategy?
A4: all the possible responses of his adversary.