Dedekind cut

Named after German mathematician Richard Dedekind (1831–1916), who introduced the concept (although a similar construction was used in Euclid's Elements to define proportional segments).

Dedekind cut (plural Dedekind cuts)

1. (mathematics) Any partition of the set of rational numbers into non-empty sets A and B such that all elements of A are less than all elements of B and A contains no greatest element; intended as a construction of a real number.
   • 1990, Judith Roitman, Introduction to Modern Set Theory, Wiley, page 70:

     More than one formal solution was offered; the one which is easiest to work with (all of the solutions are provably equivalent) is the method of Dedekind cuts.

   • 1997, Reuben Hersh, What is Mathematics, Really?, Oxford University Press, page 274:

     To identify Dedekind cuts as the sought-for "real number system," we must show that they include all the rationals and irrationals—all the numbers that can be approximated with arbitrary accuracy by rationals.

   • 2011, Alexander R. Pruss, Actuality, Possibility, and Worlds, Continuum Books, page 55:

     But there are, of course, many set theoretic ways of expressing Riemannian manifolds, just as there are many set theoretic ways of expressing real numbers (one can express them as pairs of lower and upper Dedekind cuts, or as lower Dedekind cuts, or as upper Dedekind cuts, or as equivalence classes of Cauchy sequences, and so on).