Martin's Axiom, Large Continuum and Global ${\Sigma }_n^1$-Uniformization

Abstract

We construct a generic extension of $L$ satisfying Martin's Axiom, $2^{{\aleph }_0}={\aleph }_3$, a lightface ${\Delta }_3^1$ wellorder of the reals, and ${\Sigma }_n^1$-uniformization for every $n\ge 2$ simultaneously.