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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.

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  • $\begingroup$ 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? $\endgroup$ – runway44 May 12 '20 at 5:33
  • $\begingroup$ 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. $\endgroup$ – Paramanand Singh May 12 '20 at 5:46
  • $\begingroup$ So, I gather that "unessentially different" means not really different. Am I correct? $\endgroup$ – Raj Mohan Kovilath May 12 '20 at 6:00
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The 'modern' way of saying unessentially different would be different only on a set of measure zero. If two classes which both contain an infinite number of points (i.e. an open interval) differ only by one point, then distinguishing them by any reasonable measure becomes practically impossible. It can be useful to know that one class contains a specific point and the other doesn't, but for most purposes it's irrelevant or -- unessential.

Saying "differing on a set of measure zero" is arguably more precise (since we now have to be clear about how we're choosing our measure) but is arguably less intuitive (since the mathematical definition of measure is a little more complicated that intuition suggests). Saying unessential is more intuitive ("the essence of this object is unchanged") but can produce the kind of confusion you've encountered.

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  • $\begingroup$ My confusion is now totally cleared. Thank you very much. $\endgroup$ – Raj Mohan Kovilath May 12 '20 at 8:44

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