the Cantor–Dedekind axiom is the thesis that the real numbers are order-isomorphic to the linear continuum of geometry. In other words, the axiom states...

    N2   1  4  9 16 25 36 49 64 81 100 ... Dedekind's work in this area anticipated that of Georg Cantor, who is commonly considered the founder of set...

yield results about the linear continuum of geometry relies on the Cantor–Dedekind axiom. The Greek mathematician Menaechmus solved problems and proved theorems...

During his honeymoon in the Harz mountains, Cantor spent much time in mathematical discussions with Richard Dedekind, whom he had met at Interlaken in Switzerland...

(Äquivalenzsatz). 1887 Cantor publishes the theorem, however without proof. 1887 On July 11, Dedekind proves the theorem (not relying on the axiom of choice) but...

mathematical logic, the Peano axioms (/piˈɑːnoʊ/, [peˈaːno]), also known as the Dedekind–Peano axioms or the Peano postulates, are axioms for the natural numbers...

initiated by the German mathematicians Richard Dedekind and Georg Cantor in the 1870s. In particular, Georg Cantor is commonly considered the founder of set...

fulfills some axioms of real numbers or a constructive rephrasing thereof. Various models have been studied, such as the Cauchy reals or the Dedekind reals,...

An axiom, postulate, or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments...

In mathematics, the axiom of choice, abbreviated AC or AoC, is an axiom of set theory. Informally put, the axiom of choice says that given any collection...

theorem. The modern study of set theory was initiated by Georg Cantor and Richard Dedekind in the 1870s. However, the discovery of paradoxes in naive set...

Dedekind-infinite. The axiom of countable choice (ACω), a weak variant of the axiom of choice (AC), is needed to show that a set that is not Dedekind-infinite...

determine whether or not this proof is constructive. Cantor's correspondence with Richard Dedekind shows the development of his ideas and reveals that...

number n. Any other set is infinite. Assuming the axiom of choice, it can be proved that the Dedekind notions correspond to the standard ones. It can also...

interpreted in terms of negative theology. Cantor also mentioned the idea in his letters to Richard Dedekind (text in square brackets not present in original):...

that the theory of real closed fields is decidable). Given the Cantor–Dedekind axiom, this algorithm can be regarded as an algorithm to decide the truth...

In mathematics, the axiom of determinacy (abbreviated as AD) is a possible axiom for set theory introduced by Jan Mycielski and Hugo Steinhaus in 1962...

axiom schema of specification, also known as the axiom schema of separation (Aussonderungsaxiom), subset axiom, axiom of class construction, or axiom...

characterizations can be proven equivalent in Zermelo–Fraenkel set theory without the axiom of choice, but the equivalence of the third and fourth cannot be proved...

mathematics, contains the axiom of infinity, which means that the natural numbers form a set (necessarily infinite). A great discovery of Cantor is that, if one...

the form of real products with the identity matrix in the ring. Cantor–Dedekind axiom Chronology Cuisenaire rods Extended real number line Hyperreal number...

Cantor and Richard Dedekind can be reduced to a few definitions and seven principles or axioms. He says he has not been able to prove that the axioms...

Cantor–Dedekind axiom Dedekind completeness Dedekind cut Dedekind discriminant theorem Dedekind domain Dedekind eta function Dedekind function Dedekind group Dedekind...

with Peano axioms. Secondly, both definitions involve infinite sets (Dedekind cuts and sets of the elements of a Cauchy sequence), and Cantor's set theory...

the power set of its power set is a Dedekind-infinite set, having a proper subset equinumerous to itself. If the axiom of choice is also true, then infinite...

In mathematics, the axiom of regularity (also known as the axiom of foundation) is an axiom of Zermelo–Fraenkel set theory that states that every non-empty...

definition he gave later. The resulting argument uses only five axioms of set theory. Cantor's set theory was controversial at the start, but later became...

theorem Cantor–Dedekind axiom Heine–Cantor theorem Cantor–Schröder–Bernstein theorem Cantor–Schröder–Bernstein property Smith–Volterra–Cantor set Cantor (asteroid)...

cardinality of a Dedekind-infinite set in contexts where this may not be equivalent to "infinite cardinal"; that is, in contexts where the axiom of countable...

y} have the same length, then the lattice is said to satisfy the Jordan–Dedekind chain condition. A lattice ( L , ≤ ) {\displaystyle (L,\leq )} is called...