2020/06/22 by Zurab Janelidze, Janelidze, Zurab, Ineke van der Berg +1
Computer Science · Mathematics · #03E10 #Advanced Topology and Set Theory #Category Theory (math.CT) #Computability, Logic, AI Algorithms #FOS: Mathematics #Logic (math.LO) #Mathematical and Theoretical Analysis
paper · pdf · doi:10.48550/arxiv.2006.12688
openalex publication_date 2020/06/22 · openalex created_date 2022/10/01 · openalex updated_date 2026/07/28
In this paper, we give an axiomatization of the ordinal number system, in the\nstyle of Dedekind's axiomatization of the natural number system. The latter is\nbased on a structure (N,0,s) consisting of a set N, a distinguished element\n0\∈ N and a function s colon N\→ N. The structure in our axiomatization\nis a triple (O,L,s), where O is a class, L is a function defined on all\ns-closed `subsets' of O, and s is a class function s colon O\→ O. In\nfact, we develop the theory relative to a Grothendieck-style universe (minus\nthe power-set axiom), as a way of bringing the natural and the ordinal cases\nunder one framework. We also establish a universal property for the ordinal\nnumber system, analogous to the well-known universal property for the natural\nnumber system.\n