Axiomatizations of Natural Numbers
Publications de l'Institut Mathématique, _N_S_118 (2025) no. 132, p. 25
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We introduce two axiomatizations of natural numbers and place them in the context of the well-known formalizations of natural numbers by Frege, Dedekind, Peano, Russell, and Devide. To this end, we are developing a methodology and notations that allow a uniform presentation of these different formalizations. We prove that our axiomatizations categorically axiomatize the structure $(N, 0, \pi)$, where the predecessor relation $\pi$ can be the immediate predecessor $p$ or the general predecessor $$. The first three axioms for the immediate and general predecessor are exactly the same, but the fourth axioms are specific for $p$ and $$. One postulates that the inverse of the immediate predecessor is a function, the other that the general predecessor is a total relation. We do not postulate that the inverse is an injection or that $$ is an order. Finally, we discuss Henkin's analysis of Peano's axiomatization in the same context.
Zvonimir Šikić. Axiomatizations of Natural Numbers. Publications de l'Institut Mathématique, _N_S_118 (2025) no. 132, p. 25 . doi: 10.2298/PIM2532025S
@article{10_2298_PIM2532025S,
author = {Zvonimir \v{S}iki\'c},
title = {Axiomatizations of {Natural} {Numbers}},
journal = {Publications de l'Institut Math\'ematique},
pages = {25 },
year = {2025},
volume = {_N_S_118},
number = {132},
doi = {10.2298/PIM2532025S},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.2298/PIM2532025S/}
}
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