At this time it was the custom for students in Germany to study at a number of different universities and indeed that is precisely what Zermelo did. His studies were undertaken at three universities, namely Berlin, Halle and Freiburg, and the subjects he studied were quite wide ranging and included mathematics, physics and philosophy.
At these universities he attended courses by Frobenius, Lazarus Fuchs, Planck, Schmidt, Schwarz and Edmund Husserl. This was an impressive collection of inspiring lecturers and Zermelo began to undertake research in mathematics after completing his first degree. His doctorate was completed in 1894 when the University of Berlin awarded him the degree for a dissertation Untersuchungen zur Variationsrechnung Ⓣ which followed the Weierstrass approach to the calculus of variations. In this thesis he :-
... extended Weierstrass's method for the extrema of integrals over a class of curves to the case of integrands depending on derivatives of arbitrarily high order, at the same time giving a careful definition of the notion of neighbourhood in the space of curves.After the award of his doctorate, Zermelo remained at the University of Berlin where he was appointed assistant to Planck who held the chair of theoretical physics there. At this stage Zermelo's work was turning more towards areas of applied mathematics and, under Planck's guidance, he began to work for his habilitation thesis studying hydrodynamics.
In 1897 Zermelo went to Göttingen, perhaps the leading centre for mathematical research in the world at that time, where he completed his habilitation, submitting his dissertation Hydrodynamische Untersuchungen über die Wirbelbewegungen in einer Kugelfläche Ⓣ in 1899. Immediately following the award of the degree he was appointed as a lecturer at Göttingen on the strength of his contributions to statistical mechanics as well as to the calculus of variations.
The direction of Zermelo's research was soon to take a major change. Cantor had put forward the continuum hypothesis in 1878, conjecturing that every infinite subset of the continuum is either countable (i.e. can be put in 1-1 correspondence with the natural numbers) or has the cardinality of the continuum (i.e. can be put in 1-1 correspondence with the real numbers). The importance of this was seen by Hilbert who made the continuum hypothesis the first in the list of problems which he proposed in his Paris lecture of 1900. Hilbert saw this as one of the most fundamental questions which mathematicians should attack in the 1900s and he went further in proposing a method to attack the conjecture. He suggested that first one should try to prove another of Cantor's conjectures, namely that any set can be well ordered.
Perhaps it would be helpful to give a definition of a well ordered set at this point. A set S is well ordered if it has a relation < defined on it which satisfies three properties:
(i) for any elements a, b in S either a = b, a < b or b < a.The set of natural numbers with the usual ordering is therefore a well ordered set but the set of integers is not well ordered with the usual ordering since the subset of negative integers has no least element.
(ii) for every a, b, c in S with a < b and b < c then a < c.
(iii) every non-empty subset of S has a least element.
Zermelo began to work on the problems of set theory, in particular taking up Hilbert's idea to head towards a resolution of the problem of the continuum hypothesis. In 1902 Zermelo published his first work on set theory which was on the addition of transfinite cardinals. Two years later, in 1904, he succeeded in taking the first step suggested by Hilbert towards the continuum hypothesis when he proved that every set can be well ordered. This result brought fame to Zermelo and also earned him a quick promotion for, in December 1905, he was appointed as professor in Göttingen.
The axiom of choice is the basis for Zermelo's proof that every set can be well ordered; in fact the axiom of choice is equivalent to the well ordering property so we now know that this axiom has to be used. His proof of the well ordering property used the axiom of choice to construct sets by transfinite induction. Although Zermelo certainly gained fame for his proof of the well ordering property, set theory at this time was in the rather unusual position that many mathematicians rejected the type of proofs that Zermelo had discovered. There were strong feelings as to whether such non-constructive parts of mathematics were legitimate areas for study and Zermelo's ideas were certainly not accepted by quite a number of mathematicians :-
The proof stirred the mathematical world and produced a great deal of criticism - most of it unjustified - which Zermelo answered elegantly in Neuer Beweis Ⓣ...As this quote indicates, Zermelo's reaction to these criticisms was to try to prove the well ordering property with a proof that would find more widespread acceptance, and this he succeeded in doing in the paper Neuer Beweis Ⓣ which he published in 1908. It was a paper which he specifically directed at critics of his work. On the one hand he emphasised the formal character of his new proof of the well ordering and on the other hand he argued that his critics, and other mathematicians, also used the axiom of choice when dealing with infinite sets.
Zermelo made other fundamental contributions to axiomatic set theory which were partly a consequence of the criticism of his first major contribution to the subject and partly because set theory began to become an important research topic at Göttingen. The set theory paradoxes first appeared around 1903 with the publication of Russell's paradox. Zermelo had in fact discovered a similar set paradox himself but did not publish the result. Rather it prompted him to make the first attempt to axiomatise set theory and he began this task in 1905. Having produced an axiom system he wanted to prove that his axioms were consistent before publishing the work, but he failed to achieve this.
In 1908 Zermelo published his axiomatic system despite his failure to prove consistency. He gave seven axioms : Axiom of extensionality, Axiom of elementary sets, Axiom of separation, Power set axiom, Union axiom, Axiom of choice and Axiom of infinity.
Zermelo usually stated his axioms and theorems in words rather than symbols. In fact he did not often use the formal language for quantifiers such as ∃ or ∀ and binding variables that were then being used, instead, he used ordinary expressions such as "there exists" or "for all".
It is worth commenting that Skolem and Fraenkel independently improved Zermelo's axiom system in around 1922. The resulting system, with ten axioms, is now the most commonly used one for axiomatic set theory. It enables the contradictions of set theory to be eliminated yet the results of classical set theory excluding the paradoxes can be derived.
In 1910 Zermelo left Göttingen when he was appointed to the chair of mathematics at the Zürich University. His health was poor but his position was helped by the award of a prize of 5000 marks for his major contributions to set theory. The prize was awarded on the initiative of Hilbert and certainly it was an attempt to enable Zermelo rest and so to regain his health.
When his health had not improved by 1916 Zermelo resigned his chair in Zürich and moved to the Black Forest in Germany where he lived for ten years. He was appointed to an honorary chair at Freiburg im Breisgau in 1926 but he renounced his chair in 1935 because of his disapproval of Hitler's regime. At the end of the World War II Zermelo requested that he be reinstated to his honorary position in Freiburg and indeed he was reinstated to the post in 1946.
Article by: J J O'Connor and E F Robertson
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