MT2002 Analysis

## Dedekind cuts

The first construction of the Real numbers from the Rationals is due to the German mathematician Richard Dedekind (1831 - 1916). He developed the idea first in 1858 though he did not publish it until 1872. This is what he wrote at the beginning of the article.

*As professor in the Polytechnic School in Zürich I found myself for the first time obliged to lecture upon the ideas of the differential calculus and felt more keenly than ever before the lack of a really scientific foundation for arithmetic. In discussing the notion of the approach of a variable magnitude to a fixed limiting value, and especially in proving the theorem that every magnitude which grows continuously but not beyond all limits, must certainly approach a limiting value, I had recourse to geometric evidences. ... This feeling of dissatisfaction was so overpowering that I made the fixed resolve to keep mediating on the question till I should find a purely arithmetic and perfectly rigorous foundation for the principles of infinitesimal analysis.*

He defined a real number to be a pair (*L*, *R*) of sets of rationals which have the following properties.

- Every rational is in exactly one of the sets

- Every rational in
*L* is < every rational in *R*

Such a pair is called a **Dedekind cut** (*Schnitt* in German). You can think of it as defining a real number which is the least upper bound of the "Left-hand set" *L* and also the greatest lower bound of the "right-hand set" *R*. If the cut defines a rational number then this may be in either of the two sets.

It is rather a rather long (and tedious) task to define the arithmetic operations and order relation on such cuts and to verify that they do then satisfy the axioms for the Reals -- including even the Completeness Axiom.
Richard Dedekind, along with Bernhard Riemann was the last research student of Gauss. His arithmetisation of analysis was his most important contribution to mathematics, but was not enthusiastically received by leading mathematicians of his day, notably Kronecker and Weierstrass. His ideas were, however, warmly welcomed by Jordan and especially by Cantor with whom he became firm friends.

JOC September 2001