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the definition of a well order is that if $R$ is a linear (order) and every non-empty subset of $A$ has a least element. I understand that

$(\mathbb N,\le)$ is a well-order but how come

$(I,\le)$ with subset of negative integers is not a well-order?

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Can you come up with a set of negative integers that does not have a least element?

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    The question is: given *any* set of negative integers, will it *necessarily* be true that you can find a least element? The answer is no. You could ask similar questions about whether an arbitrary set of positive integers needs to have a greatest element (the answer is no -- an exercise left to you!).2012-11-01