A Universe with large Continuum, global $\Sigma$-Uniformization and a projective Well-Order of its Reals

Abstract

We construct a model in which the continuum has size $\kappa$ for a regular cardinal $\kappa$ and in which the ${\Sigma }_n^1$-uniformization property holds simultaneously for every $n\ge 2$. Additionally this model has a ${\Delta }_3^1$- definable well-order of its reals.