Forcing the $\Pi^1_3$-Reduction Property and a Failure of $\Pi^1_3$-Uniformization

TL;DR

This paper constructs a model within ZFC where the $\Pi^1_3$-reduction property holds but the $\Pi^1_3$-uniformization property fails, demonstrating a separation between these principles.

Contribution

It shows how to force a model with the $\Pi^1_3$-reduction property while reducing the large cardinal assumptions needed, and exhibits a failure of uniformization.

Findings

The $\Pi^1_3$-reduction property can be obtained in ZFC without large cardinals.

The $\Pi^1_3$-uniformization property fails in this model.

This is the first known separation of these two principles.

Abstract

We force over the constructible universe to obtain a model of the $\Pi^1_3$-reduction property, thus lowering the best known large cardinal strength from the existence of $M_1^{\#}$ to just ZFC. In this model the $\Pi^1_3$-uniformization property fails, which separates these two principles for the first time.