Recovering a structure from its tree of tuples

Abstract

We show that for each countable ordinal $α$, there is a $\mathbfΔ^0_α$ operator that, for all $A$ of Scott rank at most $α$, takes copies of $T_A$ to copies of $A$, where $T_A$ is the tree of tuples corresponding to $A$. For finite $α$, we show that this is best possible. The proof that there is a $\mathbfΔ^0_α$ operator uses Harrington's method of workers. The sharpness result, for finite $n$, uses recent work of Harrison-Trainor on ``unfriendly jump inversion.''