Are There Numbers Beyond the Reals? I understand that reals are defined as "completing" the rationals, which (at least in ZFC) are in turn derived from the natural numbers.
So, if ordinal numbers are viewed as an extension of the natural numbers, are there constructs of ordinals corresponding in the same way as rationals and reals are derived from natural numbers ?
If there are, then does this have any serious implications for models of reality which seem to be predominantly based on the reals ?
 A: These are called Surreal Numbers. It is a proper-class sized order field which is complete, in the sense that every subset has a least upper bound and greatest lower bound.
It embeds all the ordered fields in the universe [of set theory], and all the ordinals too. It does not, however, extend ordinal arithmetic since ordinal addition is not commutative.
There are also non-standard fields which extend the real numbers, like the hyperreal numbers, and more. But these are "mere sets". So they do not extend the ordinals.

Finally, let me point out that mathematicians, in particular set theorists and model theorists don't consider their mathematics as a model of the physical universe. We keep the church and state separate from our end.
Moreover, if you consider the expansion of the universe and how every point look outwards thinks that it is in the center of the universe, wouldn't it be much more fitting to use some ultrametric space to model the universe (i.e. a space where every point inside an ball is its center) rather than Euclidean metric spaces? (For example the $p$-adic numbers seem like a good candidate for that.)
A: The complex numbers are the completion of the reals in the algebraic sense, by which I mean the set of solutions to polynomials in $\mathbb{R}[x]$ is $\mathbb{C}$.
