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Okay so for my upcoming test I need to "be able to explain at least one result that would not hold if the axiom of completeness were not accepted"

My teacher suggested that I could try to explain why Cantor's diagonalization method won't work without the axiom of completeness, but I'm not really sure why it wouldn't work?

EDIT: If you can think of any easier examples I could explain for my test, that be great as well!

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    I assume you are referring to the "complete ordered field" axiomatization of the real numbers. The rational numbers satisfy all of the other axioms, so contrasting the rational numbers to the real numbers should help you figure things out.2012-11-04
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    If this is a real analysis class, then I would rather highlight the importance of completeness in proving the intermediate value theorem.2012-11-04
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    Suppose you go through the diagonal argument and the number you find that is not on the list is $\pi-3$. Without completeness, we could say that is not a real number.2012-11-04

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