On $\boldsymbol{\Sigma}^1_3$- and $\Sigma^1_4$-uniformization

TL;DR

This paper constructs a set-theoretic model demonstrating that the $oldsymbol{ riangle}^1_3$-uniformization property can hold while the $oldsymbol{ riangle}^1_4$-uniformization property fails, highlighting a separation between these principles.

Contribution

It provides the first explicit model separating $oldsymbol{ riangle}^1_3$- and $oldsymbol{ riangle}^1_4$-uniformization properties under ZFC consistency.

Findings

Constructed a model where $oldsymbol{ riangle}^1_3$-uniformization holds but $oldsymbol{ riangle}^1_4$-fails.

Indicated how to create a universe with the same properties using inner models with large cardinals.

First separation of these uniformization principles for these levels.

Abstract

Assuming the consistency of $\mathsf{ZFC}$, we construct a model of set theory in which the boldface $\mathbf{\Sigma}^1_3$-uniformization property holds, yet the lightface $\Sigma^1_4$-uniformization property fails, separating these two principles for the first time. We also indicate how to create a universe where $\Sigma^1_3$-uniformization holds, but $\Sigma^1_4$-uniformization fails using inner models with large cardinals.