Ernst Zermelo

Ernst Friedrich Ferdinand Zermelo (July 27 1871, Berlin, German Empire – May 21 1953, Freiburg im Breisgau, West Germany) was a German mathematician, whose work has major implications for the foundations of mathematics. His best know contribution is his axiomatization of set theory called Zermelo set theory, which later leads to Zermelo-Fraenkel set theory, the set theory that is now standard. Also he is the one who first explicitly formulated the axiom of choice.

Life

Ernst Friedrich Ferdinand Zermelo was born on July 27 in 1871 at Berlin. He graduated from Berlin's Luisenstädtisches Gymnasium in 1889. He then studied mathematics, physics and philosophy at the universities of Berlin, Halle and Freiburg. He finished his doctorate in 1894 at the University of Berlin, awarded for a dissertation on the calculus of variations (Untersuchungen zur Variationsrechnung). Zermelo remained at the University of Berlin, where he was appointed assistant to Planck, under whose guidance he began to study hydrodynamics. In 1897, Zermelo went to Göttingen, at that time the leading centre for mathematical research in the world, where he completed his habilitation thesis in 1899.

In 1900, in the Paris conference of the International Congress of Mathematicians, David Hilbert challenged the mathematical community with his famous Hilbert's problems, a list of 23 unsolved fundamental questions which mathematicians should attack during the coming century. The first of these, a problem of set theory, was the continuum hypothesis introduced by Cantor in 1878.

Zermelo began to work on the problems of set theory and in 1902 published his first work concerning the addition of transfinite cardinals (a kind of numbers to describe the size of infinity). In 1904, he succeeded in taking the first step suggested by Hilbert towards the continuum hypothesis when he proved the well-ordering theorem (every set can be well ordered). This result brought fame to Zermelo, who was appointed Professor in Göttingen, in 1905. His proof of the well-ordering theorem based on the axiom of choice, the first explicit formulation of which is credit to him, was not accepted by all mathematicians, partly because set theory was not axiomatized at this time. In 1908, Zermelo succeeded in producing a much more widely-accepted proof.

In 1905, Zermelo began to axiomatize set theory; in 1908, he published his results despite his failure to prove the consistency of his axiomatic system. (See below for the details of Zermelo's axiomatization.)

In 1922, Adolf Fraenkel and Thoralf Skolem independently improved Zermelo's axiom system. The resulting 10 axiom system, now called 'Zermelo–Fraenkel set theory (ZF), is now the most commonly used system for axiomatic set theory.

In 1910, Zermelo left Göttingen upon being appointed to the chair of mathematics at Zurich University, which he resigned in 1916. He was appointed to an honorary chair at Freiburg im Breisgau in 1926, which he resigned in 1935 because he disapproved of Hitler's regime. At the end of World War II and at his request, Zermelo was reinstated to his honorary position in Freiburg. He died in 1953.

Zermelo Set Theory

Zermelo Zermelo set theory, as set out in an important paper in 1908 by Ernst Zermelo, is the ancestor of modern set theory. It bears certain differences from its descendants, which are not always understood, and are frequently misquoted. This article sets out the original axioms, with the original text (translated into English) and original numbering.

AXIOM I. Axiom of extensionality (Axiom der Bestimmtheit) "If every element of a set M is also an element of N and vice versa ... then M = N. Briefly, every set is determined by its elements".

AXIOM II. Axiom of elementary sets (Axiom der Elementarmengen) "There exists a (fictitious) set, the null set, ∅, that contains no element at all. If a is any object of the domain, there exists a set {a} containing a and only a as element. If a and b are any two objects of the domain, there always exists a set {a, b} containing as elements a and b but no object x distinct from them both." See Axiom of pairs.

AXIOM III. Axiom of separation (Axiom der Aussonderung) "Whenever the propositional function –(x) is definite for all elements of a set M, M possesses a subset M'  containing as elements precisely those elements x of M for which –(x) is true".

AXIOM IV. Axiom of the power set (Axiom der Potenzmenge) "To every set T there corresponds a set T' , the power set of T, that contains as elements precisely all subsets of T".

AXIOM V. Axiom of the union (Axiom der Vereinigung) "To every set T there corresponds a set ∪T, the union of T, that contains as elements precisely all elements of the elements of T".

AXIOM VI. Axiom of choice (Axiom der Auswahl): "If T is a set whose elements all are sets that are different from ∅ and mutually disjoint, its union ∪T includes at least one subset S₁ having one and only one element in common with each element of T".

AXIOM VII. Axiom of infinity (Axiom des Unendlichen) "There exists in the domain at least one set Z that contains the null set as an element and is so constituted that to each of its elements a there corresponds a further element of the form {a}, in other words, that with each of its elements a it also contains the corresponding set {a} as element".

Connection with standard set theory

The accepted standard for set theory is Zermelo-Fraenkel set theory. The links show where the axioms of Zermelo's theory correspond. There is no exact match for "elementary sets". (It was later shown that the singleton set could be derived from what is now called "Axiom of pairs". If a exists, a and a exist, thus {a,a} exists. By extensionality {a,a} = {a}.) The empty set axiom is already assumed by axiom of infinity, and is now included as part of it.

The axioms do not include the Axiom of regularity and Axiom of replacement. These were added as the result of work by Thoralf Skolem in 1922, based on earlier work by Adolf Fraenkel in the same year.

In the modern ZFC system, the "propositional function" referred to in the axiom of separation is interpreted as "any property definable by a first order formula with parameters". The notion of "first order formula" was not known in 1904 when Zermelo published his axiom system, and he later rejected this interpretation as being too restrictive.

In the usual cumulative hierarchy V_α of ZFC set theory (for ordinals α), any one of the sets V_α for α a limit ordinal larger than the first infinite ordinal ω forms a model of Zermelo set theory. So the consistency of Zermelo set theory is a theorem of ZFC set theory. Zermelo's axioms do not imply the existence of many infinite cardinals.

The axiom of infinity is usually now modified to assert the existence of the first infinite von Neumann ordinal $\omega$; it is interesting to observe that the original Zermelo axioms cannot prove the existence of this set, nor can the modified Zermelo axioms prove Zermelo's axiom of infinity. Zermelo's axioms (original or modified) cannot prove the existence of $V_{\omega }$ as a set nor of any rank of the cumulative hierarchy of sets with infinite index.

Bibliography

Primary literature in English translation:

• Jean van Heijenoort, 1967. From Frege to Godel: A Source Book in Mathematical Logic, 1879-1931. Harvard Univ. Press.
  • 1904. "Proof that every set can be well-ordered," 139-41.
  • 1908. "A new proof of the possibility of well-ordering," 183-98.
  • 1908. "Investigations in the foundations of set theory I," 199-215.
• 1930. "On boundary numbers and domains of sets: new investigations in the foundations of set theory" in Ewald, William B., ed., 1996. From Kant to Hilbert: A Source Book in the Foundations of Mathematics, 2 vols. Oxford Uni. Press: 1219-33.

Secondary:

• Ivor Grattan-Guinness, 2000. The Search for Mathematical Roots 1870-1940. Princeton Uni. Press.

External links

• John J. O'Connor and Edmund F. Robertson. Ernst Zermelo at the MacTutor archive
• Ernst Zermelo at the Mathematics Genealogy Project
• Scientific Biography of Ernst Zermelo (1871-1953).

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