Relatively acceptable notation

TL;DR

This paper explores the connection between Shapiro's acceptability of notations for natural numbers and computable structure theory, introducing new results on degree spectra and computability of successor functions in structures.

Contribution

It reconstructs Shapiro's approach within computable structure theory and proves a new result linking degree spectra to the computability of successor functions.

Findings

Reconstructed Shapiro's acceptability in computable structures

Proved a new relationship between degree spectra and successor computability

Provided examples illustrating the theoretical connection

Abstract

Shapiro's notations for natural numbers, and the associated desideratum of acceptability - the property of a notation that all recursive functions are computable in it - is well-known in philosophy of computing. Computable structure theory, however, although capable of fully reconstructing Shapiro's approach, seems to be off philosophers' radar. Based on the case study of natural numbers with standard order, we make initial steps to reconcile these two perspectives. First, we lay the elementary conceptual groundwork for the reconstruction of Shapiro's approach in terms of computable structures and show, on a few examples, how results pertinent to the former can inform our understanding of the latter. Secondly, we prove a new result, inspired by Shapiro's notion of acceptability, but also relevant for computable structure theory. The result explores the relationship between the classical notion of degree spectrum of a computable function on the structure in question - specifically, having all c.e. degrees as a spectrum - and our ability to compute the (image of the) successor from the (image of the) function in any computable copy of the structure. The latter property may be otherwise seen as relativized acceptability of every notation for the structure.