I know that proving a relation $\mathcal{R}$ to be a strict total order (asymmetric, transitive,and total ) implies that the relation $S$ defined as $X\mathcal{S} Y \longleftrightarrow (X\mathcal{R}Y\vee X=Y)$ is a total order (antisymmetric, transitive, reflexive, and complete).

Conversely, if a relation $\mathcal{R}$ is a total order then the relation $S$ defined as $X\mathcal{S} Y \longleftrightarrow (X\mathcal{R}Y\wedge X\neq Y)$ is a strict total order.

My perception is that people use to prove a relation to be a total order over proving that a relation is a strict total order. If proving a relation to be a total order automatically implies the other to hold why people don't do it by considering strict total orders?

For example in the definition of $\mathbb{N}$ I think it should be the same (even easier) if we prove that it's a totally strict ordered set and then automatically it's a totally order set, but books prefer the other way around. They prove that it's a totally ordered set and then the other option follows automatically. What am I missing? is it just matter of custom, style, or something?


1 Answer 1


Since I don't feel there is a lot of difference between total and partial orderings in this aspect, I will address all of them at once. It seems that the natural form of most of the orders discussed is the inclusive form.

For example, the ubiquitous subset relation $X \subseteq Y$ and its strict counterpart $X \subsetneq Y$. We express $X \subseteq Y$ by $$\forall x: x\in X \to x \in Y$$ This is a lot shorter than $X \subsetneq Y$: $$\forall x(x \in X \to x \in Y) \land \exists y( y \in Y \land \neg (y \in X))$$ which can of course be shortened by using properties of $\subseteq$, but in principle the expression above is what it comes down to.

On the other hand, the membership relation $X \in Y$ itself (considered on transitive sets such as ordinals) is (almost) never discussed in the inclusive form, because (usually) we don't want to consider sets as elements of themselves.

If you happen to know the interpretation of (inclusive) ordered sets as categories, that could be a further indication of mathematical practice: the fact that specifically inclusive ordered sets arise as special categories indicates that reasoning about them may often be "more natural" (because categories are well-behaved structures many mathematicians have a good intuition for, whether they are aware of the term or not).

Ultimately, it is, I think, not but a custom. But it is a self-reinforcing one (we use inclusive ordered sets more, so have a better intuitive grip of what their properties are. Hence they are studied more, used more in proofs (Zorn's lemma comes to mind), and appear more often in books. This all leads to positive feedback in the form of e.g. students using them more...

(I have avoided appealing to "obvious" things like "it is natural to be able to compare an element to itself" because I am probably deeply influenced by tradition and custom myself, making it "obvious".)


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