Are $x\in \mathbb{R}$ and $-\infty <x<+\infty$ indicating the same meaning in mathematics?

Question

I would like to know if it is possible in mathematics to express $x\in \mathbb{R}$ by $-\infty <x<+\infty$ in other way : Are $x\in \mathbb{R}$ and $-\infty <x<+\infty$ indicating the same meaning in mathematics ?

Thank you for any help

Answer 1

If $x$ is a real number, then the two statements are the same. The latter could also be used if $x$ were assumed to be, for example, an integer or rational number, as well as in some more exotic totally ordered sets in advanced topics.

Answer 2

Yes, these two expressions mean the same thing in every context I can imagine. (I'm sure there is some context where they mean different things - e.g. if one is considering an Archimedean ring with infinitesimal elements strictly containing $\mathbb{R}$, augmented by new elements "$+\infty$" and "$-\infty$" - but such a context would have to be specified at the outset, and I have not actually seen such a context before.) Certainly, unless explicitly specified otherwise the two expressions mean the same thing.

EDIT: A more realistic alternate use of the notation would be if $x$ were already specified to be a rational number, or an algebraic number, or an integer, or . . . Now, in these cases, $x$ is still real, so it's not quite a counterexample to what you ask, but it's worth noting.