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I was reading a book on history of numbers and came across Dedekind cut. I understand that 2 different cuts represent 2 unique numbers. But one thing that is not clear to me is if it is possible that the same Dedekind cut represent 2 different irrational numbers, i.e. 2 adjacent irrational numbers sandwiched between same cut. Can someone give proof for this?

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    There is a rational number between any two irrationals.2012-09-02
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    two different irrational numbers can't represent the same cut : you will have a rational number between them.2012-09-02
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    What does it mean for two irrational numbers to be "adjacent"?2012-09-02
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    You can't really prove that without saying what you mean by irrational number. Dedekind cuts are usually used to *define* real numbers, so that a number is identified with the cut. If you gave some other definition, it might make sense to ask if the two give the same object in some sense, but as you put it, there's nothing to prove.2012-09-02

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