are no empty locations, gaps, or discontinuities. Dedekind published his thoughts on irrational numbers and Dedekind cuts in his pamphlet "Stetigkeit und irrationale Zahlen" ("Continuity and irrational numbers"); in modern terminology, Vollständigkeit, completeness.

Dedekind defined two sets to be "similar" when there exists a one-to-one correspondence between them. He invoked similarity to give the first precise definition of an infinite set: