The Ordinals as a Consummate Abstraction of Number Systems
Alec Rhea

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 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 -fields, objects abstracted from the property that there are no two disjoint intervals in of cardinality whose union is . 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, . I proceed in this fashion until…
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Taxonomy
TopicsComputability, Logic, AI Algorithms · Advanced Topology and Set Theory · Mathematical and Theoretical Analysis
