The strong Spector-Gandy Theorem for the higher analytical pointclasses

Abstract

Assuming projective determinacy, we extend Spector's strong version of the Spector-Gandy Theorem to all odd levels of the projective hierarchy: Theorem. For every space $X$ which is a finite product of the natural numbers $N$ and Baire space $N^N$ and for every n, if $P$ is a $Π^1_{2n+1}$ subset of $X$, then there is a $Π^1_{2n}$ set $Q$ such that $P(x) \Longleftrightarrow (\exists!α)Q(x,α) \Longleftrightarrow (\existsα\inΔ^1_{2n+1}(x))Q(x,α)$.