On Sets That Encode Themselves

TL;DR

This paper explores different notions of self-encoding sets, such as introenumerable and introreducible, and introduces new methods for constructing such sets by analyzing their properties and interactions.

Contribution

It provides new results on the relationships between various self-encoding notions and introduces a novel construction method for nontrivial introenumerable and introreducible sets.

Findings

Reversal of previous results on uniformity in introenumerable sets

Construction of new nontrivial introenumerable sets

Enhanced understanding of self-encoding set properties

Abstract

Given partial information about a set, we are interested in fully recovering the original set from what is given. If a set encodes itself robustly, any partial information about the set suffices to fully recover the information about the original set. Jockusch defined a set $A$ to be introenumerable if each infinite subset of $A$ can enumerate $A$, and this is an example of a set which encodes itself. There are several other notions capturing self-encoding differently. We say $A$ is uniformly introenumerable if each infinite subset of $A$ can uniformly enumerate $A$, whereas $A$ is introreducible if each infinite subset of $A$ can compute $A$. We investigate properties of various notions of self-encoding and prove new results on their interactions. Greenberg, Harrison-Trainor, Patey, and Turetsky showed that we can always find some uniformity from an introenumerable set. We show that this can be reversed: we can construct an introenumerable set by patching up uniformity. This gives a rise to a new method of constructing a nontrivial introenumerable or introreducible set.