The strong Spector-Gandy Theorem for the higher analytical pointclasses

TL;DR

Under projective determinacy, the paper generalizes Spector's strong Spector-Gandy Theorem to all odd levels of the projective hierarchy, establishing a uniform definability result for certain sets.

Contribution

It extends the strong Spector-Gandy Theorem to all odd projective levels assuming projective determinacy, covering finite products of natural numbers and Baire space.

Findings

The theorem holds for all odd projective levels under the assumption.

Provides a uniform characterization of $ ext{Pi}^1_{2n+1}$ sets.

Connects definability with unique existential quantification over $ ext{Delta}^1_{2n+1}$ sets.

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 $\Pi^1_{2n+1}$ subset of $X$, then there is a $\Pi^1_{2n}$ set $Q$ such that $P(x) \Longleftrightarrow (\exists!\alpha)Q(x,\alpha) \Longleftrightarrow (\exists\alpha\in\Delta^1_{2n+1}(x))Q(x,\alpha)$.