Full Idea
A Dedekind Cut is a division of rationals into two set (A1,A2) where every member of A1 is less than every member of A2. If n is the largest A1 or the smallest A2, the cut is produced by n. Some cuts aren't produced by rationals.
Gist of Idea
Cuts are made by the smallest upper or largest lower number, some of them not rational
Source
Stewart Shapiro (Philosophy of Mathematics [1997], 5.4)
Book Reference
Shapiro,Stewart: 'Philosophy of Mathematics:structure and ontology' [OUP 1997], p.171