At P985 of the book, says

But there are cuts that are not determined by rational numbers. If we put into the first class all negative rational numbers and all positive ones whose squares are less than 2, and put into the second class all the other rationals, then this cut is not determined by a rational number. To each such cut "we create a new irrational member a which is fully defined by this cut; we will say that the number α corresponds to this cut or that it brings about this cut." Hence there corresponds to each cut one and only one either rational or irrational number.

The first class should also include zero, the definition of such two classes in Dedekind’s own words:   

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