# Well ordering theorem, partial ordering

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.

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?

• "… 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. – Daniel Fischer Oct 20 '16 at 12:21
• OK, sorry was a bit too quick. The subsets should of course be nonempty as you say. – JezuzStardust Oct 20 '16 at 13:43

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.