Dedekindkatkojen

Dedekindkatkojen, also known as Dedekind cuts, are a fundamental concept in real analysis used to define the real numbers. They were developed by the German mathematician Richard Dedekind in the 19th century. A Dedekind cut is a partition of the set of rational numbers into two non-empty subsets, say A and B, such that every element of A is less than every element of B. More formally, for any rational numbers x and y, if x is in A and y is in B, then x < y. Additionally, the set A must not have a greatest element, and the set B must not have a least element.

The set A is called the lower set and the set B is called the upper set. The Dedekind cut is typically represented by the pair (A, B). Dedekind used these cuts to construct the real numbers from the rational numbers. Each Dedekind cut corresponds to a unique real number. If the lower set A contains a greatest rational number, then the cut corresponds to that rational number. If A does not contain a greatest rational number, then the cut corresponds to an irrational number. The set B essentially defines the "gap" that the irrational number fills.

This construction allows for a rigorous definition of real numbers, including irrational numbers like the square root of 2 or pi, which cannot be expressed as a ratio of two integers. Dedekind cuts provide a way to formalize the completeness property of the real number line, meaning that there are no "holes" in the line. This property is crucial for many results in calculus and real analysis. The concept has since become a standard part of the axiomatic construction of the real number system.