**Ernst Zermelo** was a German mathematician who did pioneering work on the foundations of mathematics.

- 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.
- This was an impressive collection of inspiring lecturers and Zermelo began to undertake research in mathematics after completing his first degree.
- 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 Ⓣ(Hydrodynamic studies of the vortex motion in a spherical surface) in 1899.
- The direction of Zermelo's research was soon to take a major change.
- 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.
- 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.
- 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.
- 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 Ⓣ(New evidence) which he published in 1908.
- 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.
- Zermelo had in fact discovered a similar set paradox himself but did not publish the result.
- In 1908 Zermelo published his axiomatic system despite his failure to prove consistency.
- Zermelo usually stated his axioms and theorems in words rather than symbols.
- It is worth commenting that Skolem and Fraenkel independently improved Zermelo's axiom system in around 1922.
- In 1910 Zermelo left Göttingen when he was appointed to the chair of mathematics at the Zürich University.
- 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.

Born 27 July 1871, Berlin, Germany. Died 21 May 1953, Freiburg im Breisgau, Germany.

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Origin Germany, Puzzles And Problems, Set Theory

Axioms: 1 2 3

Examples: 4

Parts: 5 6

**O’Connor, John J; Robertson, Edmund F**: MacTutor History of Mathematics Archive