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At the Wikipedia page for the well ordering theorem

For every set X, there exists a well-ordering with domain X.

Furthermore, a well ordering is defined to be a strict total order so that each subset has a least element.

Now I have a couple of questions about this:

  1. Does this mean that the set X itself has a least element? After all X is a subset of itself (unless the theorem is talking about strict subsets).

  2. Some books (e.g. Manifolds and Differential Geometry, by J.M. Lee, Appendix B) defines a well ordered set to be one where there exists a partial ordering so that every subset has a least element. And then uses the well ordering theorem together with this definition.

I can easily see that the well ordering theorem implies this (since every total ordering is a partial ordering), but is there any reason why this form is stated and not the "stronger" one directly? Are they perhaps equivalent?

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    $\begingroup$ "… each nonempty subset has a least element". Ad 1, yes $X \subset X$, hence $X$ has a least element, provided $X \neq \varnothing$. Ad 2, they are equivalent. A partial order such that every nonempty subset has a least element is a total order - consider two-element subsets to see that all elements are comparable. $\endgroup$ Oct 20, 2016 at 12:21
  • $\begingroup$ OK, sorry was a bit too quick. The subsets should of course be nonempty as you say. $\endgroup$ Oct 20, 2016 at 13:43

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First of all, note that the definition requires that every non-empty subset has a least element, since the empty set is always a subset but it has no least element.

Secondly, yes, it means that $X$ has a least element, at least if $X$ is non-empty. Exactly because it is a subset of itself.

And finally, if you understand "least" as "minimum", then the answer is that the two definition are equivalent, since if $\{x,y\}$ is any two elements subset, then it has a minimum, let's say $x$, so it means that $x<y$. Therefore every two distinct elements are comparable, and the order is a total order.

If you under "least" as "minimal", then the answer is of course negative, since the empty relation is a strict partial order where every element is minimal, so every non-empty set is a minimal element.

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  • $\begingroup$ What I mean by least is that it should be less than all other elements in the subset, so I guess it is the first of the two versions that I am talking about. Is the second version, where one refers to minimal elements eve considered in this context? $\endgroup$ Oct 20, 2016 at 13:41
  • $\begingroup$ Less in the basic context, but in set theory and order theroy these partial orders play a key role. Such a partial order is called a well founded partial order. $\endgroup$
    – Asaf Karagila
    Oct 20, 2016 at 13:46

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