Why a least element is a distinct property?

Question

A Well-Ordered Set always have a least element iff it's non-empty.

Then why is this a distinct property that has a lot of attention while having a greatest element property (like $\mathbb{Z}^-$) has no attention.

Answer 1

In principle we could repeat the entire theory of well-orders with the words and relation signs flipped such that we speak about "greatest" elements of arbitrary nonempty subsets instead of "least".

Since formal logic doesn't care about the words we use, we would have exactly the same theorems in the new formulations, just in the different direction. But there are some pragmatic arguments for spending more attention on the usual concept of well-orders:

• Well-orders have an intuitively appealing main example in the form of the natural numbers. There's nothing really comparable for co-well-orders.

• It would turn the intuition about induction awry. One main use for well-orders is to construct things by long induction over the well order. Here you imagine constructing your thing starting with the first elements of the order and then your induction hypothesis tells you that there's no way you won't get a unique result for everything. Whereas the underlying formalism doesn't care about our ideas of the construction being "spread out in time" (what the induction theorem says is that the whole thing we're defining just is there, period), it would certainly make it harder to think about if we had to imagine starting with the last elements for an inductive construction.

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Note, by the way that if you combine the concepts and want to speak about ordered subsets where every nonempty subset has both a least and a greatest element, then you get a rather uninteresting theory out of it: Those orders are exactly the finite total orders. So it is essential with some asymmetry in the assumptions in order to get the rich theory of ordinals, even though it is formally irrelevant whether the asymmetry is to one side or the other.