User blog comment:Edwin Shade/Enumerating the Countable Ordinals/@comment-30754445-20171206034326/@comment-32876686-20171209002232

I looked up the definition of 'well-ordered', and I don't see how an uncountable set is not well-ordered, in that every non-empty subset has a least element.

If you were given any subset of the reals, isn't the lowest value in that subset already contained in the set ? It doesn't seem like there could be a set which doesn't have a lowest element for any given subset.