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

TL;DR

This paper constructs a model of set theory with Martin's Axiom, continuum size aleph_3, a definable wellorder of reals, and uniformization properties for all projective levels above 1.

Contribution

It introduces a method to build a generic extension of L satisfying multiple complex set-theoretic and descriptive set-theoretic properties simultaneously.

Findings

Achieves a model with MA and continuum aleph_3.

Ensures a lightface Δ^1_3 wellorder of the reals.

Provides Σ^1_n-uniformization for all n ≥ 2.

Abstract

We construct a generic extension of $L$ satisfying Martin's Axiom, $2^{\aleph_0}=\aleph_3$, a lightface $\Delta^1_3$ wellorder of the reals, and $\Sigma^1_n$-uniformization for every $n\geq 2$ simultaneously.