How is the extended real number line modeled? In set theory, numbers are often constructed, e.g. from nestings of sets which eventually contain the empty set. The operations are defined in term of taking unions etc.etc. 
The extended real number line hat plus and minus infinity $\infty,-\infty$ in addition to the reals, and they fulfill certain axioms.

How are the two objects "$\infty$" and "$-\infty$" of the extended real number line $\mathbb R\cup\{-\infty,\infty\}$ modeled in rigorous set theory? 

I figure it might be related to ordinal numbers. But these are two things and they behave somewhat similarly. So I wonder what they are made of, set theoretically.
 A: One way to do this is using the Dedekind cut construction of the real numbers from the rationals. If one relaxes the requirement that the left and right sets of a Dedekind cut must be nonempty, one has the two cuts $(\emptyset, \mathbb{Q})$ and $(\mathbb{Q}, \emptyset)$. These can be identified with $-\infty$ and $\infty$, respectively.
Then we have, for example, the following as a natural result:


*

*$\infty + \infty = \infty$ (add the elements of the left set of $\infty$, i.e. $\mathbb{Q}$ (that is, form the set $\{ x + y : x \in \mathbb{Q}, y \in \mathbb{Q} \}$, to those of itself and you get $\mathbb{Q}$ again, same goes for the right set)

*$-\infty + -\infty = -\infty$ (add the elements of the left set of $-\infty$, i.e. $\emptyset$, to those of itself and you get $\emptyset$ again, same goes for the right set)

*$a + \infty = \infty$ (for a non-infinity cut $a$, note that if you add the elements of the left set of $a$ to $\mathbb{Q}$, you just get $\mathbb{Q}$ again)


all from the usual definitions of addition of Dedekind cuts. Note that $-\infty + \infty$ doesn't work, though, since if we try that, both the left and right set collapse, which is not a valid cut. But we expect that since we usually consider $-\infty + \infty = \infty - \infty$ undefined.
A: All you need is two objects that are not in $\mathbb R$, so you may let $\infty=\{\mathbb R\}$ and $-\infty=\{\{\mathbb R\}\}$, for example.
