Computing sets from all infinite subsets

TL;DR

This paper explores the complexity and properties of introreducible sets, showing they are highly complex and that every introenumerable set contains an introreducible subset, revealing deep structural insights.

Contribution

It proves that the collection of introreducible sets is $ ext{Pi}^1_1$-complete and that every introenumerable set has an introreducible subset, advancing understanding of their complexity and structure.

Findings

Introreducible sets are $ ext{Pi}^1_1$-complete, indicating high complexity.

Every introenumerable set contains an introreducible subset.

Introreducible sets encode information robustly across all their infinite subsets.

Abstract

A set is introreducible if it can be computed by every infinite subset of itself. Such a set can be thought of as coding information very robustly. We investigate introreducible sets and related notions. Our two main results are that the collection of introreducible sets is $\Pi^1_1$-complete, so that there is no simple characterization of the introreducible sets; and that every introenumerable set has an introreducible subset.