Set theoretical Representations of Integers, I

TL;DR

This paper explores how different classical semantics of integers can be distinguished using Kolmogorov complexity, revealing a hierarchy that reflects their degree of abstraction and contrasting with traditional invariance results.

Contribution

It introduces a novel effectivization of integer semantics within Kolmogorov theory and characterizes the resulting complexity hierarchy, linking it to jump oracles and infinite computations.

Findings

Kolmogorov complexities form a hierarchy based on semantics

Hierarchy coincides with complexities defined via jump oracles

Contrasts with invariance of usual Kolmogorov complexity

Abstract

We reconsider some classical natural semantics of integers (namely iterators of functions, cardinals of sets, index of equivalence relations), in the perspective of Kolmogorov complexity. To each such semantics one can attach a simple representation of integers that we suitably effectivize in order to develop an associated Kolmogorov theory. Such effectivizations are particular instances of a general notion of "self-enumerated system" that we introduce in this paper. Our main result asserts that, with such effectivizations, Kolmogorov theory allows to quantitatively distinguish the underlying semantics. We characterize the families obtained by such effectivizations and prove that the associated Kolmogorov complexities constitute a hierarchy which coincides with that of Kolmogorov complexities defined via jump oracles and/or infinite computations. This contrasts with the well-known fact that usual Kolmogorov complexity does not depend (up to a constant) on the chosen arithmetic representation of integers, let it be in any base unary, binary et so on. Also, in a conceptual point of view, our result can be seen as a mean to measure the degree of abstraction of these diverse semantics.