On graphs of total projective functions

Abstract

It is well known that the graph of a total $\mathbfΣ^1_n$-function is $\mathbfΠ^1_n$. 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 $\mathbfΠ^1_3$-function is $\mathbfΣ^1_3$. This principle is incompatible with $\mathbfΠ^1_3$-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}.