Rigid many-one degrees contain infinite antichains of $1$-degrees
Patrizio Cintioli
Abstract
Odifreddi asked whether every non-irreducible many-one degree must contain an infinite antichain of one-one degrees. Positive answers are known for computably enumerable many-one degrees (Degtev) and, more recently, for many-one degrees admitting a $Δ^0_2$ representative (Batyrshin). In this note we isolate a rigidity principle behind these phenomena. Call a set $A\subseteqω$ \emph{$m$-rigid} if every total computable $m$-autoreduction of $A$ is eventually the identity. We prove that if $A$ is $m$-rigid, then its many-one degree $°_m(A)$ contains an infinite antichain of $1$-degrees. The proof uses a uniform duplication construction: for each computable parameter $S$ we define $B_S\equiv_m A$ so that any injective reduction $B_S\le_1 B_T$ induces an $m$-autoreduction of $A$ and therefore forces $S\subseteq^{*}T$. Choosing an almost-inclusion infinite antichain of computable sets yields the desired infinite $1$-antichain inside $°_m(A)$. As applications, Jockusch's rigidity theorem implies that every $1$-generic set is $m$-rigid, giving a comeager family of positive instances. Moreover, $m$-rigidity holds with Lebesgue measure $1$ (indeed, every Martin-Löf random real is $m$-rigid). Consequently, Odifreddi's Question~5 has a positive answer \emph{with probability $1$} for a fair-coin random $A\in 2^ω$; any counterexample (if it exists) is confined to a null set (and, by genericity, also to a meager set).