praj-17/QA-MCQ-Open-ended huggingface dataset:
Title: John_von_Neumann
Context: 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.

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

answer: limericks

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

answer: regularly playing extremely loud German march music

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

answer: Albert Einstein,
Title: John_von_Neumann
Context: 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.

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

answer: Stanislaw Ulam

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

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

answer: a new collection of jokes
Title: John_von_Neumann
Context: 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.

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

answer: Russell's paradox

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

answer: Ernst Zermelo and Abraham Fraenkel

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.
Title: John_von_Neumann
Context: 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.

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

answer: method of inner models

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

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
Title: John_von_Neumann
Context: 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.

question: What is a proper class? 

answer: a class which does not belong to other classes

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

answer: Zermelo–Fraenkel

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.
Title: John_von_Neumann
Context: 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.

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

answer: Kurt Gödel

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

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

answer: Congress of Königsberg
Title: John_von_Neumann
Context: 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."

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

answer: less than a month

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

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

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

answer: second incompleteness theorem
Title: John_von_Neumann
Context: 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.

question: What field did Von Neuman establish? 

answer: continuous geometry.

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

question: What was the first example of continuous geometry? 

answer: projections of the hyperfinite type II factor
Title: John_von_Neumann
Context: 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."

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

answer: In a series of famous papers

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

answer: algebraic concept of solvability of a group is relevant

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
Title: John_von_Neumann
Context: 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.

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

answer: methods of argument he employed

question: In 1938 what was von Neumann awarded? 

answer: Bôcher Memorial Prize

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
Title: John_von_Neumann
Context: 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.

question: What is von Neumann algebra? 

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

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.

question: When was direct integral introduced by von Neumann? 

answer: 1949

question: What was introduced in von Neumann algebras? 

answer: study of rings of operators
Title: John_von_Neumann
Context: 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".

question: What did von Neumann work on in 1937 - 39? 

answer: lattice theory

question: What properties determine dimension in lattice theory? 

answer: conserved by perspective mappings ("perspectivities") and ordered by inclusion

question: Who wrote that "John von Neumann's brilliant mind blazed over lattice theory like a meteor?" 

answer: Garrett Birkhoff
Title: John_von_Neumann
Context: 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."

question: What type of axioms were used by von Neumann in his lattice work? 

answer: novel axioms

question: How long was von Neumann's lattice theory paper? 

answer: 140 pages

question: In what room did von Neumann often write at home? 

answer: living room writing-table
Title: John_von_Neumann
Context: 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.

question: Who established the framework for quantum mechanics? 

answer: Von Neumann

question: What paper did von Neumann produce in 1932? 

answer: Mathematical Foundations of Quantum Mechanics

question: What von Neumann work preceded the axiomatiztion of quatum mechanics? 

answer: axiomatization of set theory
Title: John_von_Neumann
Context: 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?"

question: What are the physics of quantum mechanics? 

answer: mathematics of Hilbert spaces and linear operators acting on them

question: What is the uncertainty principle? 

answer: determination of the position of a particle prevents the determination of its momentum and vice versa

question: The physics of quantum mechanics included special cases for what work? 

answer: formulations of both Heisenberg and Schrödinger
Title: John_von_Neumann
Context: 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.

question: Von Neumann's abstract treatment allowed him to work on what foundational issue? 

answer: determinism versus non-determinism

question: What was John Bell's argument? 

answer: proof contained a conceptual error and was therefore invalid

question: What did Jeffrey Bub argue in 2010? 

answer: Bell had misconstrued von Neumann's proof

question: Did von Neumann rule hidden variable theories? 

answer: von Neumann did not claim that his proof completely ruled out hidden variable theories
Title: John_von_Neumann
Context: 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).

question: What results came from von Neumann's deep analysis of the measurement problem? 

answer: physical universe could be made subject to the universal wave function

question: What "outside of calculation" variable could initiate collapse in von Neumann's conclusion? 

answer: consciousness of the experimenter

question: Were von Neumann's conclusions of collapse of entire universe cause accepted widely? 

answer: Von Neumann–Wigner interpretation never gained acceptance amongst the majority of physicists
Title: John_von_Neumann
Context: 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.

question: With whom did von Neuman work on a paper in 1936 that introduce quantum logic? 

answer: Garrett Birkhoff

question: What concept was created for quatum logic? 

answer: propositional calculus

question: What is the difference of logic in quantum theory? 

answer: laws of distribution of classical logic,  and , are not valid for quantum theory
Title: John_von_Neumann
Context: 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.

question: What year was game theory established? 

answer: 1928

question: What discipline is game theory derived from? 

answer: mathematical discipline.

question: what are the possible strategies in minimax theory? 

answer: a pair of strategies for both players that allows each to minimize his maximum losses

question: What must a player consider when determining every possible strategy? 

answer: all the possible responses of his adversary.
Title: John_von_Neumann
Context: The reason for this is that a quantum disjunction, unlike the case for classical disjunction, can be true even when both of the disjuncts are false and this is, in turn, attributable to the fact that it is frequently the case, in quantum mechanics, that a pair of alternatives are semantically determinate, while each of its members are necessarily indeterminate. This latter property can be illustrated by a simple example. Suppose we are dealing with particles (such as electrons) of semi-integral spin (angular momentum) for which there are only two possible values: positive or negative. Then, a principle of indetermination establishes that the spin, relative to two different directions (e.g., x and y) results in a pair of incompatible quantities. Suppose that the state ɸ of a certain electron verifies the proposition "the spin of the electron in the x direction is positive." By the principle of indeterminacy, the value of the spin in the direction y will be completely indeterminate for ɸ. Hence, ɸ can verify neither the proposition "the spin in the direction of y is positive" nor the proposition "the spin in the direction of y is negative." Nevertheless, the disjunction of the propositions "the spin in the direction of y is positive or the spin in the direction of y is negative" must be true for ɸ. In the case of distribution, it is therefore possible to have a situation in which , while .

question: What is the difference of quantum disjunction from classic? 

answer: can be true even when both of the disjuncts are false

question: What is frequently the case in quantum mechanics when a pair of alternatives are semantically determinate? 

answer: each of its members are necessarily indeterminate

question: How can quantum disjuction be illustrated? 

answer: illustrated by a simple example