Dedekind’s postulate
Dedekind’s postulate
If K1 and K2 are any two non-empty parts of K, such that every element of K belongs either to K1 or to K2 and every element of K1 precedes every element of K2, then there is at least one element x in K such that (1) any element that precedes x belongs to K1, and (2) any element that follows x belongs to K2. Here K is a class ordered by a relation R (see order), and it is said that y precedes z, and that z follows y, if yRz and y?z. If K is densely ordered by R and in addition satisfies Dedekind’s postulate, it is said to have continuous order. — C.A.B.