A $wtt$-introimmune set in \texorpdfstring{$\Pi^0_1$}{Pi01} and introimmunity for several reducibilities

TL;DR

This paper establishes the existence and non-existence of introimmune sets within various computational hierarchies and reducibilities, advancing understanding of immunity properties in computability theory.

Contribution

It proves the existence of a $ ext{WTT}$-introimmune set in $ ext{Pi}^0_1$, and explores introimmunity for several reducibilities, delineating the boundaries within the arithmetical hierarchy.

Findings

Existence of a $ ext{WTT}$-introimmune set in $ ext{Pi}^0_1$

Existence of $ ext{Delta}^0_2$ sets that are $bs$- and $D$-introimmune

No infinite $ ext{Pi}^0_1$ set is $Q$-introimmune

Abstract

We prove that there exists a weak truth-table introimmune set in the class $\Pi^0_1$, settling the question left open in previous work of whether the known $\Delta^0_2$ existence result can be improved to $\Pi^0_1$. Since $\Sigma^0_1$ sets cannot be immune, this is best possible for weak truth-table introimmunity. We also study introimmunity for Jockusch's bounded-search reducibility $\le_{bs}$ and Andersen's Dartmouth reducibility $\le_D$, proving the existence of $\Delta^0_2$ sets that are $bs$-introimmune and $D$-introimmune; hence there also exists a $\Delta^0_2$ $D^+$-introimmune set. We next consider the classical reducibility $\le_Q$, which is not contained in $\le_T$ on all subsets of $\omega$. We show that no infinite $\Pi^0_1$ set is $Q$-introimmune, while a $\Delta^0_2$ $Q$-introimmune set does exist. Thus the existence of $\Delta^0_2$ $Q$-introimmune sets is best possible within the arithmetical hierarchy. Finally, for enumeration reducibility $\le_e$, we show that no infinite $\Pi^1_1$ set is $e$-introimmune, although $e$-introimmune sets do exist in the unrestricted sense. The proofs combine finite-injury priority arguments with dynamic spacing methods for $\le_{wtt}$, $\le_{bs}$, and $\le_D$, a bit-by-bit finite-extension construction for $\le_Q$, and an application of Soare's abstract existence theorem in the enumeration case.