# Constructing Wadge classes

## TL;DR

Under the Axiom of Determinacy, the paper demonstrates that all non-selfdual Wadge classes can be systematically constructed from level ω₁ classes using expansion and separated differences, providing new insights and proofs in descriptive set theory.

## Contribution

It introduces a method to construct all non-selfdual Wadge classes from level ω₁ classes through specific operations, offering a new proof of Van Wesep's theorem.

## Key findings

- Every non-selfdual Wadge class can be constructed from level ω₁ classes.
- A new proof of Van Wesep's theorem is provided.
- The construction uses expansion and separated differences operations.

## Abstract

We show that, assuming the Axiom of Determinacy, every non-selfdual Wadge class can be constructed by starting with those of level $\omega_1$ (that is, the ones that are closed under Borel preimages) and iteratively applying the operations of expansion and separated differences. The proof is essentially due to Louveau, and it yields at the same time a new proof of a theorem of Van Wesep (namely, that every non-selfdual Wadge class can be expressed as the result of a Hausdorff operation applied to the open sets). The exposition is self-contained, except for facts from classical descriptive set theory.