Abstract
The first part of this paper is a presentation of some common applications of ordinals: definition of a system of ordinal notations for ordinals less than Γ0, direct connection between Kruskal's theorem and Γ0, consistency proofs in proof theory (such as the consistency of Peano arithmetic by means of transfinite induction up to ε0). In the second part of the paper, a functorial construction of ordinals and in particular of the Veblen hierarchy is explained. This approach, introduced by Girard (theory of dilators), allows the construction of ordinals greater than Γ0 to be pursued in a more natural way than if the Bachmann hierarchy is used.
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Vauzeilles, J. Ordinals II: Some applications and a functorial approach. Ann Math Artif Intell 16, 27–57 (1996). https://doi.org/10.1007/BF02127793
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DOI: https://doi.org/10.1007/BF02127793