On Sets That Encode Themselves

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.