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?


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

  • $\begingroup$ This is exactly the part where it gets me. I don't really completely understand the "a least element" segment For example, -2 is less than -1 and so on $\endgroup$ – Aaron Nov 1 '12 at 5:38
  • $\begingroup$ With the positive integers, I could take any subset and find a least element. For example, if I take the set of all even positive integers, there is a least element: $2$. What if you take the set of all even negative integers? Is there a least one? (What if you take the whole set of negative integers; is there a least negative integer?) $\endgroup$ – Benjamin Dickman Nov 1 '12 at 5:40
  • $\begingroup$ There is infinite, well I should not say Infinite, more like None. Okay, It makes sense now. Do I have the right concept? Makes more sense if i say "arbitrary" $\endgroup$ – Aaron Nov 1 '12 at 5:43
  • $\begingroup$ 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!). $\endgroup$ – Benjamin Dickman Nov 1 '12 at 5:52

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