TITLE: How is the extended real number line modeled?
QUESTION [6 upvotes]: 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.

REPLY [7 votes]: 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.