A Dedekind-style axiomatization and the corresponding universal property of an ordinal number system. (English) Zbl 07620695
Summary: In this paper, we give an axiomatization of the ordinal number system, in the style of Dedekind’s axiomatization of the natural number system. The latter is based on a structure \((N,0,s)\) consisting of a set \(N\), a distinguished element \(0\in N\) and a function \(s:N\to N\). The structure in our axiomatization is a triple \((O,L,s)\), where \(O\) is a class, \(L\) is a class function defined on all \(s\)-closed ‘subsets’ of \(O\), and \(s\) is a class function \(s\colon O\to O\). In fact, we develop the theory relative to a Grothendieck-style universe (minus the power set axiom), as a way of bringing the natural and the ordinal cases under one framework. We also establish a universal property for the ordinal number system, analogous to the well-known universal property for the natural number system.
MSC:
| 03E10 | Ordinal and cardinal numbers |
| 03E70 | Nonclassical and second-order set theories |
| 03E45 | Inner models, including constructibility, ordinal definability, and core models |
| 11U99 | Connections of number theory and logic |
| 06A05 | Total orders |
| 06A15 | Galois correspondences, closure operators (in relation to ordered sets) |
| 54F05 | Linearly ordered topological spaces, generalized ordered spaces, and partially ordered spaces |
| 18A40 | Adjoint functors (universal constructions, reflective subcategories, Kan extensions, etc.) |
| 08A65 | Infinitary algebras |
Keywords:
class; counting system; Dedekind-Peano axioms; Grothendieck universe; initial object; limit function; ordinal number; ordinal number system; universal property; specialization preorder; well-ordered set; successor functionReferences:
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