Clarification about real number axioms

Question

Similar in spirit to a question here, why do we say as part of the axioms for real numbers that: $$\text{"The order relation $<$ has the least upper bound property."}$$ when (presumably) we mean: $$\text{"The set of real numbers $\mathbb{R}$ equipped with the usual order $<$ has the least upper bound property."}$$

To me, it seems wrong or maybe slightly misleading to say the order relation itself has the l.u.b. property since the order relation is a subset of $\mathbb{R}^2$ in this case, or a Cartesian product generally.

Is this reasoning correct, and is the original statement essentially an abuse of terminology?

Answer 1

The second statement is implicit in the first.

The first statement is neither wrong nor misleading. Saying that a relation has a property implicitly refers to the set on which the relation is defined.

I would not call it an abuse of terminology.

If you are more comfortable making the set explicit, do that when you write, or fill it in when you read.

Answer 2

In your context, the order relation of course is a binary relation, so indeed is a subset of $\mathbb{R}^2$. Then, $(\mathbb{R},<)$ has the least upper bound property. It's just a short form to say what you stated in the second centered line.

If we were studying abstract algebra or topology, for instance, where we use less usual sets, and different kind of orders, we clarify it.