Originally Posted by

**Timsworth** However, is my understanding of a greatest/maximal element with respect to arbitrary orderings correct (and then dually for <-least)?

That is, with the example of <-greatest being 0 rather than nothing (infinity), because it's the one element that holds for all other elements by transitivity. The next greatest would be 1, then 2 etc.

I suppose it's another way of asking, or even pondering whether the requirement for a set to be well-ordered is that its elements range over the domain of the natural numbers. Can a set of names be well-ordered if it's ordered lexicographically? e.g. is {adder,bird,cat,dog} well-ordered, since all non-empty subsets appear to have a 'least' element in terms of spelling? Are we relying on an intuitive notion of 'least' rather than any specific general definition?