Cantor–Dedekind axiom

The Cantor–Dedekind axiom states that points on a line correspond bijectively and order-preservingly to real numbers. It identifies the line’s geometric continuity with the real numbers’ numerical continuity.

Connect

Real number line: A representation of real numbers as points on a line, ordered by their numerical values. It is the geometric picture the axiom makes exact.
Real number: An element of the complete ordered field extending the rational numbers. These are the numerical values assigned to line points.
Richard Dedekind: A German mathematician who developed foundational accounts of real numbers and arithmetic. His construction of real numbers clarified how numerical continuity can model a line.
Rational line: The ordered set of rational numbers represented as points on a line, containing gaps relative to the real continuum. It lacks the completeness required for the full correspondence asserted by the axiom.
Continuum (mathematics): A continuous mathematical structure with no gaps, often modeled by the real numbers. The axiom asserts that a geometric line has this numerical continuity.
Rational number: A number expressible as a ratio of two integers with nonzero denominator. Rational coordinates alone leave gaps that the axiom’s real coordinates fill.
Georg Cantor: A German mathematician who founded set theory and studied infinite sets and real numbers. His work on continuity and the real numbers helped shape the principle’s formulation.
Cauchy sequence: A sequence whose terms become arbitrarily close to one another as the indices increase. Completion by Cauchy sequences offers another construction of the real numbers behind the line model.
Dedekind cut: A partition of the rational numbers into lower and upper sets that defines a real number. Cuts give a precise construction of the numerical continuum identified with the line.
Order completeness: The property that every nonempty bounded-above subset of an ordered field has a least upper bound. Completeness distinguishes the real line from the rational line.