"One of his remarkable abilities was his power of absolute recall. As far as I could tell, von Neumann was able on once reading a book or article to quote it back verbatim; moreover, he could do it years later without hesitation. He could also translate it at no diminution in speed from its original language into English. On one occasion I tested his ability by asking him to tell me how A Tale of Two Cities started. Whereupon, without any pause, he immediately began to recite the first chapter and continued until asked to stop after about ten or fifteen minutes." Excerpt, The Computer from Pascal to von Neumann by Herman Goldstein (1980)

An unconventional parent, von Neumann’s father Max would reportedly bring his workaday banking decisions home to the family and ask his children how they would have reacted to particular investment possibilities and balance-sheet risks (Macrae, 1992). He was home-schooled until 1914, as was the custom in Hungary at the time. Starting at the age of 11, he was enrolled in the German-speaking Lutheran Gymnasium in Budapest. He would attend the high school until 1921, famously overlapping the high school years of three other “Martians” of Hungary:

Leo Szilard (att. 1908–16 at Real Gymnasium), the physicist who conceived of the nuclear chain reaction and in late 1939 wrote the famous Einstein-Szilard letter for Franklin D. Roosevelt that resulted in the formation of the Manhattan Project that built the first atomic bomb

Eugene Wigner (att. 1913–21 at Lutheran Gymnasium), the 1963 Nobel Prize laureate in Physics who worked on the Manhattan Project, including the theory of the atomic nucleus, elementary particles and Wigner’s Theorem in quantum mechanics

Edward Teller (att. 1918–26 at Minta School), the “father of the hydrogen bomb”, an early member of the Manhattan Project and contributor to nuclear and molecular physics, spectroscopy and surface physics

Although all of similar ages and interests, as Macrae (1992) writes:

"The four Budapesters were as different as four men from similar backgrounds could be. They resembled one another only in the power of the intellects and in the nature of their professional careers. Wigner [...] is shy, painfully modest, quiet. Teller, after a lifetime of successful controversy, is emotional, extroverted and not one to hide his candle. Szilard was passionate, oblique, engagé, and infuriating. Johnny [...] was none of these. Johnny's most usual motivation was to try to make the next minute the most productive one for whatever intellectual business he had in mind." - Excerpt, John von Neumann by Norman Macrae (1992)

Yet still, the four would work together off and on as they all emigrated to America and got involved in the Manhattan Project.

By the time von Neumann enrolled in university in 1921, he had already written a paper with one of his tutors, Mikhail Fekete on “A generalization of Fejér’s theorem on the location of the roots of a certain kind of polynomial” (Ulam, 1958). Fekete had along with Laszló Rátz reportedly taken a notice to von Neumann and begun tutoring him in university-level mathematics. According to Ulam, even at the age of 18, von Neumann was already recognized as a full-fledged mathematician. Of an early set theory paper written by a 16 year old von Neumann, Abraham Fraenkel (of Zermelo-Fraenkel set theory fame) himself later stated (Ulam, 1958):

Letter from Abraham Fraenkel to Stanislaw Ulam

Around 1922-23, being then professor at Marburg University, I received from Professor Erhard Schmidt, Berlin [...] a long manuscript of an author unknown to me, Johann von Neumann, with the title Die Axiomatisierung der Mengerlehre, this being his eventual doctor dissertation which appeared in the Zeitschrift only in 1928 [...] I asked to express my view since it seemed incomprehensible. I don't maintain that I understood anything, but enough to see that this was an outstanding work, and to recognize ex ungue leonem [the claws of the lion]. While answering in this sense, I invited the young scholar to visit me in Marburg, and discussed things with him, strongly advising him to prepare the ground for the understanding of so technical an essay by a more informal essay which could stress the new access to the problem and its fundamental consequences. He wrote such an essay under the title Eine Axiomatisierung der Mengerlehre and I published it in 1925.

In University (1921–1926)

As Macrae (1992) writes, there was never much doubt that Johnny would one day be attending university. Johnny’s father, Max, initially wanted him to follow in his footsteps and become a well-paid financier, worrying about the financial stability of a career in mathematics. However, with the help of the encouragement from Hungarian mathematicians such as Lipót Fejér and Rudolf Ortvay, his father eventually acquiesced and decided to let von Neumann pursue his passions, financing his studies abroad.

Johnny, apparently in agreement with his father, decided initially to pursue a career in chemical engineering. As he didn’t have any knowledge of chemistry, it was arranged that he could take a two-year non-degree course in chemistry at the University of Berlin. He did, from 1921 to 1923, afterwards sitting for and passing the entrance exam to the prestigious ETH Zurich. Still interested in pursuing mathematics, he also simultaneously entered University Pázmány Péter (now Eötvös Loránd University) in Budapest as a Ph.D. candidate in mathematics. His Ph.D. thesis, officially written under the supervision of Fejér, regarded the axiomatization of Cantor’s set theory. As he was officially in Berlin studying chemistry, he completed his Ph.D. largely in absentia, only appearing at the University in Budapest at the end of each term for exams. While in Berlin, he collaborated with Erhard Schmidt on set theory and also attended courses in physics, including statistical mechanics taught by Albert Einstein. At ETH, starting in 1923, he continued both his studies in chemistry and his research in mathematics.

“Evidently, a Ph.D. thesis and examinations did not constitute an appreciable effort” — Eugene Wigner

Two portraits of John von Neumann (1920s)

In mathematics, he first studied Hilbert’s theory of consistency with German mathematician Hermann Weyl. He eventually graduated both as a chemical engineer from ETH and with Ph.D. in mathematics, summa cum laude from the University of Budapest in 1926 at 24 years old.

“There was a seminar for advanced students in Zürich that I was teaching and von Neumann was in the class. I came to a certain theorem, and I said it is not proved and it may be difficult. von Neumann didn’t say anything but after five minutes he raised his hand. When I called on him he went to the blackboard and proceeded to write down the proof. After that I was afraid of von Neumann” — George Pólya

From von Neumann’s Fellowship application to the International Education Board (1926)

His application to the Rockefeller-financed International Education Board (above) for a six-month fellowship to continue his research at the University of Göttingen mentions Hungarian, German, English, French and Italian as spoken languages, and was accompanied by letters of recommendation from Richard Courant, Hermann Weyl and David Hilbert, three of the world’s foremost mathematicians at the time (Leonard, 2010).

In Göttingen (1926–1930)

The Auditorium Maximum at the University of Göttingen, 1935

Johnny traveled to Göttingen in the fall of 1926 to continue his work in mathematics under David Hilbert, likely the world’s foremost mathematician of that time. Reportedly, according to Leonard (2010), von Neumann was initially attracted to Hilbert’s stance in the debate over so-called metamathematics, also known as formalism and that this is what drove him to study under Hilbert. In particular, in his fellowship application, he wrote of his wish to conduct (Leonard, 2010)

"Research over the bases of mathematics and of the general theory of sets, especially Hilbert's theory of uncontradictoriness [...], [investigations which] have the purpose of clearing up the nature of antinomies of the general theory of sets, and thereby to securely establish the classical foundations of mathematics. Such research render it possible to explain critically the doubts which have arisen in mathematics"

Very much both in the vein and language of Hilbert, von Neumann was likely referring to the fundamental questions posed by Georg Cantor regarding the nature of infinite sets starting in the 1880s. von Neumann, along with Wilhelm Ackermann and Paul Bernays would eventually become Hilbert’s key assistants in the elaboration of his Entscheidungsproblem (“decision problem”) initiated in 1918. By the time he arrived in Göttingen, von Neumann was already well acquainted with the topic, in addition to his Ph.D. dissertation having already published two related papers while at ETH.

Set theory

John von Neumann wrote a cluster of papers on set theory and logic while in his twenties:

von Neumann (1923). His first set theory paper is entitled Zur Einführung der transfiniten Zahlen (“On the introduction of transfinite numbers”) and regards Cantor’s 1897 definition of ordinal numbers as order types of well-ordered sets. In the paper, von Neumann introduces a new theory of ordinal numbers, which regards an ordinal as the set of the preceding ordinals (Van Heijenoort, 1970).

His first set theory paper is entitled Zur Einführung der transfiniten Zahlen (“On the introduction of transfinite numbers”) and regards Cantor’s 1897 definition of ordinal numbers as order types of well-ordered sets. In the paper, von Neumann introduces a new theory of ordinal numbers, which regards an ordinal as the set of the preceding ordinals (Van Heijenoort, 1970). von Neumann (1925) . His second set theory paper is entitled Eine Axiomatisierung der Mengenlehre (“An axiomatization of set theory”). It is the first paper that introduces what would later be known as the von Neumann-Bernays-Gödel set theory (NBG) and includes the first introduction of the concept of a class, defined using the primitive notions of functions and arguments. In the paper, von Neumann takes a stance in the foundations of mathematics debate, objecting to Brouwer and Weyl’s willingness to ‘sacrifice much of mathematics and set theory’, and logicists’ ‘attempts to build mathematics on the axiom of reducibility’. Instead, he argued for the axiomatic approach of Zermelo and Fraenkel, which, in von Neumann’s view, replaced vagueness with rigor (Leonard, 2010).

. His second set theory paper is entitled Eine Axiomatisierung der Mengenlehre (“An axiomatization of set theory”). It is the first paper that introduces what would later be known as the von Neumann-Bernays-Gödel set theory (NBG) and includes the first introduction of the concept of a class, defined using the primitive notions of functions and arguments. In the paper, von Neumann takes a stance in the foundations of mathematics debate, objecting to Brouwer and Weyl’s willingness to ‘sacrifice much of mathematics and set theory’, and logicists’ ‘attempts to build mathematics on the axiom of reducibility’. Instead, he argued for the axiomatic approach of Zermelo and Fraenkel, which, in von Neumann’s view, replaced vagueness with rigor (Leonard, 2010). von Neumann (1926) . His third paper Az általános nalmazelmélet axiomatikus folépitése, his doctoral dissertation, which contains the main points which would be published in German for the first time in his fifth paper.

. His third paper Az általános nalmazelmélet axiomatikus folépitése, his doctoral dissertation, which contains the main points which would be published in German for the first time in his fifth paper. von Neumann (1928) . In his fourth set theory paper, entitled Die Axiomatisierung der Mengenlehre (“The Axiomatization of Set Theory”), von Neumann formally lays out his own axiomatic system. With its single page of axioms, it was the most succinct set theory axioms developed at the time, and formed the basis for the system later developed by Gödel and Berneys.

. In his fourth set theory paper, entitled Die Axiomatisierung der Mengenlehre (“The Axiomatization of Set Theory”), von Neumann formally lays out his own axiomatic system. With its single page of axioms, it was the most succinct set theory axioms developed at the time, and formed the basis for the system later developed by Gödel and Berneys. von Neumann (1928) . His fifth paper on set theory, “Über die Definition durch transfinite Induktion und verwandte Fragen der allgemeinen Mengenlehre” (“On the Definition by Transfinite Induction and related questions of General Set Theory”) proves the possibility of definition by transfinite induction. That is, in the paper von Neumann demonstrates the significance of axioms for the elimination of the paradoxes of set theory, proving that a set does not lead to contradictions if and only if its cardinality is not the same as the cardinality of all sets, which implies the axiom of choice (Leonard, 2010).

. His fifth paper on set theory, “Über die Definition durch transfinite Induktion und verwandte Fragen der allgemeinen Mengenlehre” (“On the Definition by Transfinite Induction and related questions of General Set Theory”) proves the possibility of definition by transfinite induction. That is, in the paper von Neumann demonstrates the significance of axioms for the elimination of the paradoxes of set theory, proving that a set does not lead to contradictions if and only if its cardinality is not the same as the cardinality of all sets, which implies the axiom of choice (Leonard, 2010). von Neumann (1929). In his sixth set theory paper, Über eine Widerspruchsfreiheitsfrage in der axiomatischen Mengenlehre, von Neumann discusses the questions of relative consistency in set theory (Van Heijenoort, 1970).

Summarized, von Neumann’s main contribution to set theory is what would become the von Neumann-Bernays-Gödel set theory (NBG), an axiomatic set theory that is considered a conservative extension of the accepted Zermelo-Fraenkel set theory (ZFC). It introduced the notion of class (a collection of sets defined by a formula whose quantifiers range only over sets) and can define classes that are larger than sets, such as the class of all sets and the class of all ordinal numbers.