Punctually Standard and Nonstandard Models of Natural Numbers

TL;DR

This paper explores how different models of natural numbers with nonstandard successor functions affect the class of primitive recursive functions, identifying conditions for models to preserve standard computational properties.

Contribution

It introduces the concept of bases for punctual standardness and characterizes which operation sets preserve standard primitive recursive functions across models.

Findings

Many natural operation sets do not form bases for punctual standardness.

Finite bases for punctual standardness are identified.

Results establish punctual categoricity for certain structures.

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.