During his honeymoon in the Harz mountains, Cantor spent much time in mathematical discussions with Richard Dedekind, whom he had met two years earlier while on Swiss holiday.

That the algebra of the real numbers can be employed to yield results about the linear continuum of geometry relies on the Cantor – Dedekind axiom.

Cantor also mentioned the idea in his letters to Richard Dedekind ( text in square brackets not present in original ):

In his professorial doctoral dissertation, On the Concept of Number ( 1886 ) and in his Philosophy of Arithmetic ( 1891 ), Husserl sought, by employing Brentano's descriptive psychology, to define the natural numbers in a way that advanced the methods and techniques of Karl Weierstrass, Richard Dedekind, Georg Cantor, Gottlob Frege, and other contemporary mathematicians.

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

At that Congress, Cantor renewed his friendship and correspondence with Dedekind.

Owing to the gigantic simultaneous efforts of Frege, Dedekind and Cantor, the infinite was set on a throne and revelled in its total triumph.

As Cantor and Dedekind were developing more abstract versions of Stevin's continuum, Paul du Bois-Reymond wrote a series of papers on infinitesimal-enriched continua based on growth rates of functions.

In the second half of the nineteenth century, the calculus was reformulated by Augustin-Louis Cauchy, Bernard Bolzano, Karl Weierstrass, Cantor, Dedekind, and others using the ( ε, δ )- definition of limit and set theory.

Dedekind, Riemann, Moritz Cantor and Alfred Enneper, although they had all already earned their PhDs, attended Dirichlet's classes to study with him.

* the various ( but equivalent ) constructions of the real numbers by Dedekind and Cantor resulting in the modern axiomatic definition of the real number field ;

The first three, due to Georg Cantor / Charles Méray, Richard Dedekind and Karl Weierstrass / Otto Stolz all occurred within a few years of each other.

Bolzano's posthumously published work Paradoxien des Unendlichen ( The Paradoxes of the Infinite ) was greatly admired by many of the eminent logicians who came after him, including Charles Sanders Peirce, Georg Cantor, and Richard Dedekind.

There are also papers on Dedekind, Cantor, and Russell.

Given the Cantor – Dedekind axiom, this algorithm can be regarded as an algorithm to decide the truth of any statement in Euclidean geometry.

Mistakes and philosophical weaknesses in reasoning about infinitesimal numbers in the work of Gottfried Leibniz, Johann Bernoulli, Leonhard Euler, Augustin-Louis Cauchy, and others were the reason that they were originally abandoned for the more cumbersome real number-based arguments developed by Georg Cantor, Richard Dedekind, and Karl Weierstrass, which were perceived as being more rigorous by Weierstrass's followers.

Of a totally different orientation < nowiki > the " Old Formalist School " of Richard Dedekind | Dedekind, Georg Cantor | Cantor, Giuseppe Peano | Peano, Ernst Zermelo | Zermelo, and Louis Couturat | Couturat, etc .< nowiki ></ nowiki > was the Pre-Intuitionist School, mainly led by Poincaré, Borel and Lebesgue.

In mathematical logic, the phrase Cantor – Dedekind axiom has been used to describe the thesis that the real numbers are order-isomorphic to the linear continuum of geometry.

In the nineteenth century authors on the general theory included Laplace, Sylvestre Lacroix ( 1816 ), Littrow ( 1833 ), Adolphe Quetelet ( 1853 ), Richard Dedekind ( 1860 ), Helmert ( 1872 ), Hermann Laurent ( 1873 ), Liagre, Didion, and Karl Pearson.

The use of multisets in mathematics predates the name " multiset " by nearly 90 years: Richard Dedekind used multisets in a paper published in 1888.

In another paper in 1897, Dedekind studied the lattice of divisors with gcd and lcm as operations, so that the lattice order is given by divisibility.

In the same paper, Dedekind observed further that any lattice of ideals of a commutative ring satisfies the following stronger form of the modular identity, which is also self-dual:

A paper published by Dedekind in 1900 had lattices as its central topic: He described the free modular lattice generated by three elements, a lattice with 28 elements.

Somewhat later again Dedekind formulated the concept of ideal, Kummer having worked in a different way.

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.

The Dedekind eta function, named after Richard Dedekind, is a function defined on the upper half-plane of complex numbers, where the imaginary part is positive.

Note that where is the Dedekind eta function.

More generally, the PBW theorem as formulated above extends to cases such as where ( 1 ) L is a flat K-module, ( 2 ) L is torsion-free as an abelian group, ( 3 ) L is a direct sum of cyclic modules ( or all its localizations at prime ideals of K have this property ), or ( 4 ) K is a Dedekind domain.

In mathematics, the Dedekind zeta function of an algebraic number field K, generally denoted ζ < sub > K </ sub >( s ), is a generalization of the Riemann zeta function — which is obtained by specializing to the case where K is the rational numbers Q.

Analogously to the Riemann zeta function, the values of the Dedekind zeta function at integers encode ( at least conjecturally ) important arithmetic data of the field K. For example, the analytic class number formula relates the residue at s = 1 to the class number h ( K ) of K, the regulator R ( K ) of K, the number w ( K ) of roots of unity in K, the absolute discriminant of K, and the number of real and complex places of K. Another example is at s = 0 where it has a zero whose order r is equal to the rank of the unit group of O < sub > K </ sub > and the leading term is given by

where with and is the Dedekind eta function and the function is a holomorphic cusp form of weight 12 and level 1, known as the discriminant modular form.

Dedekind gave at least two proofs of Fermat's theorem on sums of two squares, both using the arithmetical properties of the Gaussian integers, which are numbers of the form a + bi, where a and b are integers, and i is the square root of − 1.

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

In the first section, Conway provides an axiomatic construction of numbers and ordinal arithmetic, namely, the integers, reals, the countable infinity, and entire towers of infinite ordinals, using a notation that is essentially an almost trite ( but critically important ) variation of the Dedekind cut.

The number of different Sperner families is counted by the Dedekind numbers, the first few of which numbers are

The number of different Sperner families on a set of n elements is counted by the Dedekind numbers, the first few of which are

The normal basis theorem implies that the first cohomology group of the additive group of L will vanish ; this is a result on general field extensions, but was known in some form to Richard Dedekind.

Let K be an algebraic number field K. Its Dedekind zeta function is first defined for complex numbers s with real part Re ( s ) > 1 by the Dirichlet series

** If the set A is infinite, then there exists an injection from the natural numbers N to A ( see Dedekind infinite ).

The real numbers are uniquely picked out ( up to isomorphism ) by the properties of a Dedekind complete ordered field, meaning that any nonempty set of real numbers with an upper bound has a least upper bound.

In mathematical logic, the Peano axioms, also known as the Dedekind – Peano axioms or the Peano postulates, are a set of axioms for the natural numbers presented by the 19th century Italian mathematician Giuseppe Peano.

More generally, a Dedekind cut is a partition of a totally ordered set into two non-empty parts ( A and B ), such that A is closed downwards ( meaning that for all a in A, x ≤ a implies that x is in A as well ) and B is closed upwards, and A contains no greatest element.

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 ".

If the ordered set S is complete, then, for every Dedekind cut ( A, B ) of S, the set B must have a minimal element b,

Dedekind then constructed the real numbers by Dedekind cuts, in which irrational numbers are formally defined, which serve to fill the " gaps " between rational numbers, thereby creating a complete set: the continuum of real numbers, which had already been developed by Simon Stevin in terms of decimal expansions.

However, if R is the ring of algebraic integers in an algebraic number field, or more generally a Dedekind domain, the multiplication defined above turns the set of fractional ideal classes into an abelian group, the ideal class group of R. The group property of existence of inverse elements follows easily from the fact that, in a Dedekind domain, every non-zero ideal ( except R ) is a product of prime ideals.

That is, there is a Dedekind infinite set A such that the cardinality of A is m.

In 1871, Richard Dedekind, called a set of real or complex numbers which is closed under the four arithmetic operations a " field ".

For convenience we may take the lower set as the representative of any given Dedekind cut, since completely determines.

* We form the set of real numbers as the set of all Dedekind cuts of, and define a total ordering on the real numbers as follows:

In valuation theory, the ramification theory of valuations studies the set of extensions of a valuation of a field K to an extension field of K. This generalizes the notions in algebraic number theory, local fields, and Dedekind domains.

* In the category of partially ordered sets and monotonic functions between posets, the complete lattices form the injective objects for order-embeddings, and the Dedekind – MacNeille completion of a partially ordered set is its injective hull.

By analogy with the Dedekind – MacNeille completion of a partially ordered set, every partially ordered set can be extended uniquely to a minimal chain-complete poset.