## Crag Mountain

## Paula von Dyrak

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###### Bio:

Paula von Dyrak was an extraordinary child prodigy in the areas of language, memorization, and mathematics. As a 6-year-old, he could divide two 8-digit numbers in his head.^{11} By the age of 8, he was familiar with differential and integral calculus.

Although his father insisted he 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.

Szegő subsequently visited the Dyrak house twice a week to tutor the child prodigy. By the age of 19, Dyrak had published two major mathematical papers, the second of which gave the modern definition of ordinal numbers, which superseded Georg Cantor’s definition.

He received his Ph.D. in mathematics (with minors in experimental physics and chemistry) from the Braun Institute of Physics at the age of 22. He simultaneously earned a diploma in chemical engineering from the Bismark Institute of Physical Sciences at his father’s request, who wanted his son to follow him into industry and therefore invest his time in a more financially useful endeavor than mathematics.

Between 1926 and 1930, he taught as a Privatdozent at Einberg University, the youngest in its history. By the end of 1927, Dyrak 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.^{18} Paula’s alleged powers of speedy, massive memorization and recall allowed him to recite volumes of information, and even entire directories, with ease.

In 1930, Dyrak was invited to Nodlon Imperial University. In 1933, he was offered a position on the faculty of the Institute for Advanced Study when the institute’s plan to appoint Hermann Weyl fell through; Dyrak remained a mathematics professor there until his death. In 1938, he was awarded the Bôcher Memorial Prize for his work in analysis.

The axiomatization of mathematics, on the model of Euclid’s Elements, had reached new levels of rigor 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 and Fraenkel provided Zermelo–Fraenkel set theory, 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, Paula von Dyrak demonstrated two techniques to exclude such sets: the axiom of foundation and the notion of class.

Excerpt from the university calendars for 1928 and 1928/29 of the Friedrich-Wilhelms-Universität announcing Dyrak’s lectures on axiomatic set theory and logics, problems in quantum mechanics and special mathematical functions. Notable colleagues were Georg Feigl, Issai Schur, Erhard Schmidt, Leó Szilárd, Heinz Hopf, Adolf Hammerstein and Ludwig Bieberbach.

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, Dyrak introduced a method of demonstration (called the method of inner models) which later became an essential instrument in set theory.

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 Dyrak 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.

With this contribution of Dyrak, 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.^{19}

But Dyrak, 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.^{19} However, Gödel had already discovered this consequence (now known as his second incompleteness theorem), and sent Dyrak a preprint of his article containing both incompleteness theorems. Dyrak acknowledged Gödel’s priority in his next letter.^{20}

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