Higher-Order Categoricity: A Foundational Reassessment of First-Order Axiomatics
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2025
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| _version_ | 1866902334814027776 |
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| author | SÉRGIO DE ANDRADE, PAULO |
| author_facet | SÉRGIO DE ANDRADE, PAULO |
| contents | This paper conducts a foundational reassessment of first-order axiomatics by contrasting it with the model-theoretic properties of higher-order logics, particularly second-order logic. First-order logic, characterized by the completeness and compactness theorems, is inherently unable to ensure categorical axiomatizations of fundamental infinite structures, such as the natural numbers or the real numbers. This limitation, demonstrated by the Löwenheim-Skolem theorems, gives rise to non-standard models that are structurally divergent from their intended archetypes. In contrast, second-order logic, by allowing quantification over properties and relations, can categorically define these structures, ensuring that all models are isomorphic. For example, the second-order Peano axioms and the axioms for a complete ordered field uniquely determine the natural numbers and the real numbers, respectively. However, this expressive power comes at the cost of sacrificing completeness; there is no effective proof system that can capture all second-order logical truths. This paper argues that the traditional preference for first-order logic, based on its proof-theoretic tractability, overlooks the profound semantic and descriptive advantages of higher-order logic. By examining the trade-off between deductive completeness and descriptive power, we contend that for foundational purposes where the primary goal is to characterize a unique mathematical structure, the categoricity afforded by higher-order logic is a more crucial desideratum. We conclude that a re-evaluation of the role of higher-order logic is necessary for a more faithful and robust foundation for mathematics. |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_17690628 |
| institution | Zenodo |
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| publishDate | 2025 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | Higher-Order Categoricity: A Foundational Reassessment of First-Order Axiomatics SÉRGIO DE ANDRADE, PAULO This paper conducts a foundational reassessment of first-order axiomatics by contrasting it with the model-theoretic properties of higher-order logics, particularly second-order logic. First-order logic, characterized by the completeness and compactness theorems, is inherently unable to ensure categorical axiomatizations of fundamental infinite structures, such as the natural numbers or the real numbers. This limitation, demonstrated by the Löwenheim-Skolem theorems, gives rise to non-standard models that are structurally divergent from their intended archetypes. In contrast, second-order logic, by allowing quantification over properties and relations, can categorically define these structures, ensuring that all models are isomorphic. For example, the second-order Peano axioms and the axioms for a complete ordered field uniquely determine the natural numbers and the real numbers, respectively. However, this expressive power comes at the cost of sacrificing completeness; there is no effective proof system that can capture all second-order logical truths. This paper argues that the traditional preference for first-order logic, based on its proof-theoretic tractability, overlooks the profound semantic and descriptive advantages of higher-order logic. By examining the trade-off between deductive completeness and descriptive power, we contend that for foundational purposes where the primary goal is to characterize a unique mathematical structure, the categoricity afforded by higher-order logic is a more crucial desideratum. We conclude that a re-evaluation of the role of higher-order logic is necessary for a more faithful and robust foundation for mathematics. |
| title | Higher-Order Categoricity: A Foundational Reassessment of First-Order Axiomatics |
| url | https://doi.org/10.5281/zenodo.17690628 |