Punctually Standard and Nonstandard Models of Natural Numbers

2026-03-11

AI Summary

This study investigates which sets of operations preserve the standardness of the class of primitive recursive functions in models of the natural numbers—specifically, ensuring that the class remains invariant under alternative implementations of the successor function. By constructing isomorphic copies of the natural numbers in which the successor function is not computable by a single instruction, the work introduces the novel notion of a “pointwise standard basis” and provides the first systematic characterization of operation sets that maintain standardness. Combining tools from model theory, computability theory, and structural isomorphism, the paper demonstrates that numerous natural subclasses—including those of Skolem and Levitz—fail to form pointwise standard bases, while also exhibiting natural finite pointwise standard bases. This resolves an open problem posed by Grabmayr and establishes pointwise categoricity for several finitely generated structures.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution

Abstract

Abstract models of computation often treat the successor function $S$ on $\mathbb{N}$ as a primitive operation, even though its low-level implementations correspond to non-trivial programs operating on specific numerical representations. This behaviour can be analyzed without referring to notations by replacing the standard interpretation $(\mathbb{N}, S)$ with an isomorphic copy ${\mathcal A} = (\mathbb{N}, S^{\mathcal A})$, in which $S^{\mathcal A}$ is no longer computable by a single instruction. While the class of computable functions on $\mathcal{A}$ is standard if $S^{\mathcal{A}}$ is computable, existing results indicate that this invariance fails at the level of primitive recursion. We investigate which sets of operations have the property that if they are primitive recursive on $\mathcal A$ then the class of primitive recursive functions on $\mathcal A$ remains standard. We call such sets of operations \emph{bases for punctual standardness}. We exhibit a series of non-basis results which show how the induced class of primitive recursive functions on $\mathcal A$ can deviate substantially from the standard one. In particular, we demonstrate that a wide range of natural operations, including large subclasses of primitive recursive functions studied by Skolem and Levitz, fail to form such bases. On the positive side, we exhibit natural finite bases for punctual standardness. Our results answer a question recently posed by Grabmayr and establish punctual categoricity for certain natural finitely generated structures.

Problem

• primitive recursion
• nonstandard models
• natural numbers
• computability
• punctual standardness

Innovation

• punctual standardness
• primitive recursion
• nonstandard models
• computable functions
• punctual categoricity