Relatively acceptable notation
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 wellknown 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.
 Publication:

arXiv eprints
 Pub Date:
 May 2022
 DOI:
 10.48550/arXiv.2205.00791
 arXiv:
 arXiv:2205.00791
 Bibcode:
 2022arXiv220500791B
 Keywords:

 Mathematics  Logic;
 Computer Science  Other Computer Science;
 03D45;
 03A99
 EPrint:
 15 pages, 0 figures