The subTuring degrees

TL;DR

This paper introduces subTuring reducibility for partial functions on natural numbers, revealing a complex lattice structure of realizability subtoposes of the effective topos with novel properties.

Contribution

It defines subTuring degrees and demonstrates their structure as a dense, non-modular lattice, highlighting new insights into realizability subtoposes.

Findings

SubTuring degrees form a dense, non-modular lattice.

Existence of a nonzero join-irreducible subTuring degree.

Realizability subtoposes have complex lattice properties.

Abstract

In this article, we introduce a notion of reducibility for partial functions on the natural numbers, which we call subTuring reducibility. One important aspect is that the subTuring degrees correspond to the structure of the realizability subtoposes of the effective topos. We show that the subTuring degrees (that is, the realizability subtoposes of the effective topos) form a dense non-modular (thus, non-distributive) lattice. We also show that there is a nonzero join-irreducible subTuring degree (which implies that there is a realizability subtopos of the effective topos that cannot be decomposed into two smaller realizability subtoposes).