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The first definition of well-ordering given on Proofwiki.org is:

Let (S,⪯) be an ordered set.

Then the ordering ⪯ is a well-ordering on S iff every non-empty subset of S has a smallest element under ⪯:

∀T⊆S:∃a∈T:∀x∈T:a⪯x

But the formal statement here doesn't seem right. Where is the condition that T must be non-empty? I thought there should be something between ∀T⊆S and the rest, e.g.

∀T⊆S:T≠∅→∃a∈T:∀x∈T:a⪯x

or

∀T⊆S:∀a∈T:∃b∈T:∀x∈T:b⪯x

Is the original statement wrong, or am I missing something?

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The definition in words is correct. Proofwiki's definition in symbols is incorrect, as the emptyset always fails to have an $a$, so no sets can satisfy the definition given in symbols.

The first proposed alternative definition in symbols is correct. The second is a strange but correct phrasing.

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  • $\begingroup$ Could you expand a bit more on the last comment? I thought that in the case T=∅, ∃b∈T:∀x∈T:b⪯x would still be vacuously true for all a∈T, because there is no such a. $\endgroup$ – user287393 Oct 31 '15 at 22:25
  • $\begingroup$ Okay, you're right, I've corrected the solution. $\endgroup$ – vadim123 Oct 31 '15 at 22:28

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