Dedekind cut

plural Dedekind cuts

1. mathematicsAny 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.

   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.

   1990, Judith Roitman, Introduction to Modern Set Theory, Wiley, page 70

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   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.

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

   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).

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