THE FAILURE OF SELMAN’S THEOREM FOR HYPERENUMERATION REDUCIBILITY

Abstract Hyperenumeration reducibility was first introduced by Sanchis [11]. The relationship between hyperenumeration and hyperarithmetic reducibility shares many parallels with the relationship between enumeration and Turing reducibility. We ask if this relationship can be pushed to prove and analog of Selman’s Theorem for hyperenumeration reducibility. By studying e-pointed trees in Baire space we are able to get a counter example. An e-pointed tree T is a tree with no dead ends and the property that every path in T enumerates T . We prove that if T is an e-pointed tree then for all X if T is $\Pi ^1_1$ in X then $\overline {T}$ is $\Pi ^1_1$ in X . We build an e-pointed tree T such that $\overline {T}$ is not hyperenumeration reducible to T .