Continuity, Axioms of

axioms that in various ways express the continuity of a line. For example, the Dedekind cut postulate states that if all points of a line are divided into two nonempty classes with all the points of the first class to the left of all the points of the second class, then there exists a rightmost point of the first class or a leftmost point of the second class. The Cantor axiom states that the intersection of a nested sequence of segments whose lengths tend to zero consists of a single point. The axioms of continuity make it possible to establish an order-preserving one-to-one correspondence between the set of all points on a line and the set of all real numbers.

D. Hilbert proposed as axioms of continuity the Archimedean axiom and the axiom of linear completeness. The latter axiom asserts the impossibility of adding new points to a line in such a way that the axioms of order and congruence and the Archimedean axiom are preserved.