  In the course of many mathematical developments involving 'number systems'
like $\mathbb{N}, \mathbb{Z}, \mathbb{Q}, \mathbb{R}, \mathbb {C}$ etc., it
sometimes becomes necessary to abstract away and study certain properties of
the number system in question so that we may better understand objects having
these properties in a more general setting -- a relevant example for this paper
would be Hausdorffs $\eta_\varepsilon$-fields, objects abstracted from the
property that there are no two disjoint intervals in $\mathbb{R}$ of
cardinality $\alpha \leq \aleph_0$ whose union is $\mathbb{R}$.
  My goal will be to reverse this process, so to speak -- I will begin in one
of the most abstract mathematical settings possible, using only the undefined
notions and axioms of MK class theory to define and subsequently add structure
to the class of all ordinals, $O_n$. I proceed in this fashion until the
construction (or a subclass thereof) is structurally isomorphic to whichever
'number system' we wish to consider. In the course of doing so, I construct a
proper class of set-sized non-isomorphic discretely ordered rings and a proper
class-sized discretely ordered ring, a proper class of non-isomorphic set-sized
densely ordered fields that are not real-closed and a proper class sized
densely ordered field that is not real-closed, a proper class of non-isomorphic
real-closed fields, and a proper class of non-isomorphic algebraically closed
fields. I also propose a new definition for the Surreal and Surcomplex numbers,
using a 'final culmination' of the process used to construct real-closed fields
and their algebraic closures.
  This process yields a straightforward and logically satisfying method for
moving back and forth between well-studied number systems like $\mathbb{R}$ and
very abstract number systems, such as an $\eta_{_{\omega^\omega}}$-field.

Logic