On a local variant of the 12th Delfino problem -- the $\Sigma$-side

Abstract

Assuming that $M_n$, the canonical inner model with $n$ Woodin cardinals, exists, we force a model in which every $\boldsymbol{\Sigma}^1_{n+2}$ set is Lebesgue measurable and has the Baire property, and in which $\Sigma^1_{n+2+m}$-uniformization holds for every $m\in\omega$. Additionally, this universe has a $\Delta^1_{n+3}$-definable wellorder of the reals. This answers a question of S. D. Friedman and R. Schindler from 1999. In the case $n=1$, the construction also gives a model with one Woodin cardinal in which all $\Sigma^1_3$ sets are measurable with respect to the random, Cohen, Sacks and Miller notions of measurability, while a $\Delta^1_4$-definable wellorder of the reals exists answering an instance of a question of S. D. Friedman and D. Schrittesser.