# Dedekind cut - Meaning of “unessentially different”.

In his pamphlet "Continuity and Irrational Number", in section IV, "Creation of Irrational Numbers", Dedekind compares two cuts $$(A_1,A_2)$$ and $$(B_1,B_2)$$. He then considers the case when the two classes $$A_1$$ and $$B_1$$ are not identical, and, $$a_1$$ is the only element in $$A_1$$ that is not contained in $$B_1$$. In this he proves that the cut $$(A_1,A_2)$$ and $$(B_1,B_2)$$ are produced by the same rational number. Then he says, "The two cuts are then only unessentially different". What does "unessentially different" mean?

Thanks.

• The whole point of cuts is to represent numbers. If two different cuts represent the same number, then they're not really different in a way that matters are they? – runway44 May 12 '20 at 5:33
• When defining a cut $(A, B)$ of rationals Dedekind observes three exclusive and exhaustive possibilities : lower set $A$ has a least member, upper set $B$ has a least member, neither $A$ has a greatest member nor $B$ has a least member. The first two cases are un-essentially different as every rational can be represented by both these types of cuts. Modern approach forbids the first possibility while defining a cut and thereby avoids this ambiguity using standardization/convention. – Paramanand Singh May 12 '20 at 5:46
• So, I gather that "unessentially different" means not really different. Am I correct? – Raj Mohan Kovilath May 12 '20 at 6:00