DEDEKIND , (JULIUS WILHELM) RICHARD

The son of an academic lawyer from Braunschweig, Germany, Dedekind was educated at the Caroline College there and at Göttingen, where he gained his doctorate in 1852. After four years spent teaching at Göttingen, he was appointed professor of mathematics at the Zürich Polytechnic. In 1862 he returned to Braunschweig to the Technical High School where he remained until his retirement in 1912.

In 1872 Dedekind published his most important work Stetigkeit und Irrationale Zahlen (Continuity and Irrational Numbers) in which he provided a rigorous definition of the irrational numbers. He began by ‘cutting’ or dividing the rational numbers into two non-empty disjoint sets A and B such that if x belongs to A and y toB, then x łe; y.If A has a greatest member A' then A' is a rational number; if B has a smallest number B' then B' will also be a rational number. But if A has no greatest number and B no smallest, then the cut defines an irrational number.

In the same work Dedekind gave the first precise definition of an infinite set. A set is infinite, he argued, when it is “similar to a proper part of itself.” Thus the set N of natural numbers can be shown to be ‘similar’, that is, matched or put into a one-to-one correspondence with a proper part, in this case 2N:

Whereas the only thing a finite set can be matched with is the set itself.

In a later work, Was sind und was sollen die Zahlen? (What numbers are and should be, 1888) Dedekind demonstrated how arithmetic could be derived from a set of axioms. A simpler, but equivalent version, formulated by Peano in 1889, is much better known.