Dedekind in a sentence

In 1858, Dedekind proposed a definition of the real numbers in terms of Dedekind cuts of rational numbers (Dedekind 1872), a definition still employed in contemporary texts.

In 1882, the mathematical correspondence between Cantor and Richard Dedekind came to an end, apparently as a result of Dedekind's declining the chair at Halle.

A field norm is a Dedekind-Hasse norm; thus, (5) shows that a Euclidean domain is a PID.

Also, a Dedekind domain is a UFD if and only if its ideal class group is trivial.

An 1872 meeting between Cantor and Richard Dedekind influenced Cantor's thinking and culminated in Cantor's 1874 paper.

Another way to phrase the pigeonhole principle for finite sets is similar to the principle that finite sets are Dedekind finite : Let A and B be finite sets.

Any Dedekind-complete ordered field is isomorphic to the real numbers.

Around the same time Richard Dedekind showed that the natural numbers are uniquely characterized by their induction properties.

But Dedekind did not use the term "ring" and did not define the concept of a ring in a general setting.

Dedekind domains that are not fields (for example, discrete valuation rings ) have dimension one.

Dedekind finite naturally means that every injective self-map is also surjective.

Dedekind supported it, but delayed its publication due to Kronecker's opposition.

Dedekind used the German word Schnitt main (cut) in a visual sense rooted in Euclidean geometry.

For this process, elements of the poset are mapped to (Dedekind-) cuts, which can then be mapped to the underlying posets of arbitrary complete lattices in much the same way as done for sets and free complete (semi-) lattices above.

History Ideals were first proposed by Richard Dedekind in 1876 in the third edition of his book Vorlesungen über Zahlentheorie (English: Lectures on Number Theory).

History Recursive definitions had been used more or less formally in mathematics before, but the construction of primitive recursion is traced back to Richard Dedekind 's theorem 126 of his Was sind und was sollen die Zahlen?

In other words, the number line where every real number is defined as a Dedekind cut of rationals is a complete continuum without any further gaps.

It can be a simplification, in terms of notation if nothing more, to concentrate on one "half" — say, the lower one — and call any downward closed set A without greatest element a "Dedekind cut".

It is straightforward to show that a Dedekind cut among the real numbers is uniquely defined by the corresponding cut among the rational numbers.

It was only published twelve years later in 1868 by Dedekind, two years after his death.