This entry focuses on the axiomatisation; a further entry will consider later axiomatisations of set theory in the period —, including Zermelo's second axiomatisation of The introduction to Zermelo's paper makes it clear that set theory is regarded as a fundamental theory:. Zermelo goes on:.
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View two larger pictures. His father was a college professor so Zermelo was brought up in a family where academic pursuits were encouraged. 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. This was an impressive collection of inspiring lecturers and Zermelo began to undertake research in mathematics after completing his first degree. In this thesis he [ 1 ] 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. The direction of Zermelo's research was soon to take a major change. Cantor had put forward the continuum hypothesis in , conjecturing that every infinite subset of the continuum is either countable i.
Zermelo set theory sometimes denoted by Z - , as set out in an important paper in 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. The axioms of Zermelo set theory are stated for objects, some of which but not necessarily all are called sets, and the remaining objects are urelements and do not contain any elements. Later versions of set theory often assume that all objects are sets so there are no urelements and there is no need for the unary predicate. The most widely used and accepted set theory is known as ZFC, which consists of Zermelo—Fraenkel set theory with the addition of the axiom of choice. The links show where the axioms of Zermelo's theory correspond.
He is known for his role in developing Zermelo—Fraenkel axiomatic set theory and his proof of the well-ordering theorem. He then studied mathematics , physics and philosophy at the University of Berlin , the University of Halle , and the University of Freiburg. He finished his doctorate in 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. He was appointed to an honorary chair at the University of Freiburg in , which he resigned in because he disapproved of Adolf Hitler 's regime. In , 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.