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 ${\mathbf{\Sigma }}_{n+2}^1$ set is Lebesgue measurable and has the Baire property, and in which ${\Sigma }_{n+2+m}^1$-uniformization holds for every $m\in \omega$. Additionally, this universe has a ${\Delta }_{n+3}^1$-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 ${\Sigma }_3^1$ sets are measurable with respect to the random, Cohen, Sacks and Miller notions of measurability, while a ${\Delta }_4^1$-definable wellorder of the reals exists answering an instance of a question of S. D. Friedman and D. Schrittesser.

70 pages