The Ordinals as a Consummate Abstraction of Number Systems

Por • 29 jun, 2017 • Sección: Filosofía

Alec Joshua Rhea

In the course of many mathematical developments involving ‘number systems’ like N,Z,Q,R,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 η ε   -fields, objects abstracted from the property that there are no two disjoint intervals in R  of cardinality α≤ℵ 0   whose union is 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 doing so, I hope to obtain a straightforward and logically satisfying method for moving back and forth between well-studied number systems like R  and very abstract number systems, such as a η  ω ω     -field. I also propose a new definition for the Surreal numbers, using a ‘final culmination’ of the process used to construct η ε   -fields.

arXiv:1706.08908v1 [math.LO]

 Logic (math.LO)

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