The Ordinals as a Consummate Abstraction of Number Systems

TL;DR

This paper develops a method to reconstruct various number systems from the class of all ordinals using set-theoretic axioms, creating a unified framework for understanding and relating different number systems.

Contribution

It introduces a novel process to derive classical and abstract number systems from the ordinals within MK class theory, including new definitions for Surreal and Surcomplex numbers.

Findings

Constructs a proper class of non-isomorphic discretely ordered rings.

Builds a proper class of non-isomorphic non-real-closed dense fields.

Proposes new definitions for Surreal and Surcomplex numbers.

Abstract

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.