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This is the definition of well-ordered set in analysis I of Tao,

Let X be a partially ordered set, and let Y be a totally ordered subset of X. We say that Y is well-ordered if every non-empty subset of Y has a minimal element min(Y ).

As I understand, when the condition is meet, the set Y is called well-ordered. However, if I am given a well-ordered set, say Z, I cannot say anything about it's properties, due to the implication in the definition. I look for the definition of well-ordered set on google, and found several ones that say well-ordered set is equivalent to having such properties mentioned in the definition.

What do I miss here? Is there an exception to use implication in a definition, that allows us to understand it not as usual?

Update: the implication in the definition I mentioned is, If Y be a totally ordered, and every non-empty subset of Y has a minimal element min(Y ), then Y is well-ordered.

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  • $\begingroup$ One thing that this definition lets you do is essentially induction: take the least element of the whole set, delete it, take the least element of that, delete it, and so on; repeating this process indefinitely, you reconstruct either a copy of $\{ 1,\dots,n \}$ (because the set was finite so eventually it became empty) or $\mathbb{N}$. An interesting thing is that it's possible to have a well-ordered set that is still not empty even after infinitely many such deletions. $\endgroup$
    – Ian
    Commented Aug 21, 2019 at 1:22
  • $\begingroup$ "I cannot say anything about its properties, due to the implication in the definition." Which implication? What is preventing you from saying anything about the properties of $Z$? $\endgroup$
    – littleO
    Commented Aug 21, 2019 at 1:22
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    $\begingroup$ Oh, actually I think I understand your problem: you are reading the implication as if it is one way. We usually understand definitions as being equivalences: "X has the property I am defining now if and only if X has these already-defined properties". Arguably you might understand a definition instead as "if X has the property I am defining now, then X has these already-defined properties". But there is no point in understanding definitions like the converse of that, because then "X has the property I am defining now" would be vacuous, which wouldn't help us communicate. $\endgroup$
    – Ian
    Commented Aug 21, 2019 at 1:25
  • $\begingroup$ If it helps, the second sentence is incorrect. It should be "We say that Y is well-ordered if every non-empty subset Z of Y has a minimal element min(Z)." $\endgroup$
    – user694818
    Commented Aug 21, 2019 at 1:32

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What Tao really means is,

We say that $Y$ is well-ordered if and only if every non-empty subset of $Y$ has a minimal element $\min(Y)$.

In other words, Tao defines the sentence "$Y$ is well-ordered" as meaning "every non-empty subset of $Y$ has a minimal element $\min(Y)$."

As you suspect, it's common for math writers to use "if" in a definition, when maybe "if and only if" would be more appropriate.

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  • $\begingroup$ Should I read the definition as "Y is well-ordered if and only if Y be a totally ordered, and every non-empty subset of Y has a minimal element min(Y )"? $\endgroup$
    – Jesse
    Commented Aug 21, 2019 at 1:56

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