On graphs of total projective functions

TL;DR

This paper proves the consistency of a dual assertion in descriptive set theory, showing models where the graphs of certain complex functions have opposite definability properties, challenging standard assumptions.

Contribution

It establishes the consistency of a dual statement at the third projective level, contrasting with classical results and addressing uniformization issues.

Findings

Constructs a model where all total a1_3 functions have graphs that are a0_3.

Shows this principle is incompatible with a1_3-uniformization.

Addresses the failure-of-uniformization argument from prior work.

Abstract

It is well known that the graph of a total $\mathbf{\Sigma}^1_n$-function is $\mathbf{\Pi}^1_n$. We prove the consistency of the dual assertion at the third projective level: there is a model of $\ZFC$ in which the graph of every total $\mathbf{\Pi}^1_3$-function is $\mathbf{\Sigma}^1_3$. This principle is incompatible with $\mathbf{\Pi}^1_3$-uniformization and hence with the usual projective-determinacy picture. The construction also repairs the final step of the failure-of-uniformization argument from~\cite{HOFFELNER2023103292}.