Peano's Infinite Mirrors: Automorphism Groups of Non-Standard Models
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2025
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| author | SÉRGIO DE ANDRADE, PAULO |
| author_facet | SÉRGIO DE ANDRADE, PAULO |
| contents | This paper explores the intricate structure of automorphism groups of non-standard models of Peano Arithmetic (PA), conceptualizing them as "Peano's Infinite Mirrors". Non-standard models, rich and complex structures extending the natural numbers, possess symmetries that are often far more elaborate than those found in the standard model. We delve into the foundational aspects of non-standard models, their existence, and the crucial role of model theory in their study. The core objective is to analyze the algebraic properties and combinatorial characteristics of their automorphism groups, examining how these groups reflect the internal structure and undefinability within the models. Special attention is given to the impact of model-theoretic properties such as recursiveness, saturation, and elementarity on the nature of these automorphisms. We investigate results concerning the size, rigidity, and transitive actions of these groups, drawing connections between the logical properties of PA and the structural symmetries of its non-standard realizations. The paper concludes by discussing the profound implications of these findings for our understanding of arithmetic, set theory, and the philosophy of mathematics, highlighting the ongoing challenges and open questions in this fascinating domain. |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_17689467 |
| institution | Zenodo |
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| publishDate | 2025 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | Peano's Infinite Mirrors: Automorphism Groups of Non-Standard Models SÉRGIO DE ANDRADE, PAULO This paper explores the intricate structure of automorphism groups of non-standard models of Peano Arithmetic (PA), conceptualizing them as "Peano's Infinite Mirrors". Non-standard models, rich and complex structures extending the natural numbers, possess symmetries that are often far more elaborate than those found in the standard model. We delve into the foundational aspects of non-standard models, their existence, and the crucial role of model theory in their study. The core objective is to analyze the algebraic properties and combinatorial characteristics of their automorphism groups, examining how these groups reflect the internal structure and undefinability within the models. Special attention is given to the impact of model-theoretic properties such as recursiveness, saturation, and elementarity on the nature of these automorphisms. We investigate results concerning the size, rigidity, and transitive actions of these groups, drawing connections between the logical properties of PA and the structural symmetries of its non-standard realizations. The paper concludes by discussing the profound implications of these findings for our understanding of arithmetic, set theory, and the philosophy of mathematics, highlighting the ongoing challenges and open questions in this fascinating domain. |
| title | Peano's Infinite Mirrors: Automorphism Groups of Non-Standard Models |
| url | https://doi.org/10.5281/zenodo.17689467 |