On Sets That Encode Themselves

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.

💡 Research Summary

The paper investigates a family of “self‑encoding” infinite sets—sets from which the whole can be recovered from any infinite piece of information. Starting from Jockusch’s original notions, a set A is introenumerable if every infinite subset C⊆A can enumerate A, and introreducible if every infinite subset can compute A. The authors introduce uniform versions: uniformly introenumerable (a single enumeration operator works for all infinite subsets) and uniformly introreducible (a single Turing functional works for all infinite subsets).

A central theme is the relationship between these notions and the existence of uniformity. Greenberg, Harrison‑Trainor, Patey, and Turetsky (GHTPT) proved that any introenumerable set A contains an infinite subset C and an enumeration operator Γ such that Γ enumerates A relative to every infinite subset of C. This shows that uniformity can be extracted from a non‑uniform self‑encoding set. The present work reverses this direction: given uniformity (an infinite set together with a uniform operator), one can “patch” it to build a new introenumerable set. The construction uses a forcing framework reminiscent of Hechler forcing: conditions are finite approximations to a total majorizing function together with finite strings respecting a regressive or retracing tree. By meeting a countable collection of dense sets, a sufficiently generic filter yields a total function f and a set A such that every infinite subset of A is forced to be enumeration‑above A via the same operator. Consequently, the authors provide a clean, uniformity‑based method for producing non‑trivial introenumerable (and consequently introreducible) sets, avoiding the intricate combinatorial tricks previously required.

The paper also studies regressive and retraceable sets, which are defined via a “regressing” function φ (for regressive sets) or a “retracing” function (for retraceable sets) that steps through the elements of the set. When such a set is co‑immune (has no infinite c.e. subset), the associated regressing/retracing tree is finitely branching with a unique infinite path—namely the set itself. This structural property enables a 0‑1‑oracle to effectively enumerate or compute the set, establishing that co‑immune regressive sets are 0‑1‑computable and co‑immune retraceable sets are computable. The authors extend McLaughlin’s theorem in this direction.

In Section 2 the authors connect major‑enumerable and major‑reducible notions to hyperarithmeticity. They prove that a set is Π^1_1 (or Σ^1_1) iff it is major‑enumerable (or major‑reducible) via some total function f, and that every major‑reducible set possesses a uniform modulus. As a corollary, every major‑enumerable set is Π^1_1 and therefore hyperarithmetic. Using Solovay’s theorem, they show that every major‑enumerable set has a uniformly major‑reducible subset, and conversely any set with a Δ^0_1 subset has a uniformly major‑reducible subset.

Section 3 establishes a lattice of inclusion and “refinability” relations among the eight principal notions considered: uniformly major‑enumerable, major‑enumerable, uniformly intro‑reducible, intro‑reducible, regressive, retraceable, c.e., and computable. The authors prove that all are pairwise distinct, and that any two of the four groups {major‑enumerable, uniformly major‑enumerable, regressive, retraceable} are refinable to each other (each set in one class contains an infinite subset belonging to the other). They also give explicit constructions showing the strictness of each inclusion, e.g., a uniformly major‑enumerable set that is not introreducible, a uniformly major‑reducible set that is not regressive, and a retraceable set that is not major‑enumerable.

Finally, the paper addresses several open questions from Jockusch’s original work and from GHTPT, providing answers or partial progress. Notably, it resolves Question 4.17 (in the original numbering) by demonstrating the reverse uniformity construction, and it raises new questions about the possible existence of introenumerable sets without any uniformly intro‑reducible subset, or about the exact strength of the uniformity‑patching method in higher recursion‑theoretic hierarchies.

Overall, the article clarifies the landscape of self‑encoding sets, establishes precise connections between uniform and non‑uniform notions, and introduces a powerful forcing‑based technique for constructing new introenumerable and introreducible sets. These contributions deepen our understanding of how partial information can encode complete computational content and open avenues for further exploration in computability theory and reverse mathematics.