Iterated Priority Arguments in Descriptive Set Theory

TL;DR

This paper introduces the true stages machinery and demonstrates its utility in providing new proofs for fundamental theorems in descriptive set theory, enhancing understanding of the structure of definable sets.

Contribution

The paper develops the true stages machinery and applies it to give new proofs of key theorems in descriptive set theory, offering novel insights and methods.

Findings

New proofs of Hausdorff-Kuratowski and Wadge theorems

Simplified proofs of Louveau and Saint-Raymond's separation theorem

Enhanced understanding of the structure of ${\mathbf \Delta}^0_\xi$ sets

Abstract

We present the true stages machinery and illustrate its applications to descriptive set theory. We use this machinery to provide new proofs of the Hausdorff-Kuratowski and Wadge theorems on the structure of ${\mathbf \Delta}^0_\xi$, Louveau and Saint-Raymond's separation theorem, and Louveau's separation theorem.