Omega-powers and descriptive set theory

TL;DR

This paper explores the complexity of sets of infinite sentences generated from dictionaries over finite alphabets using descriptive set theory, revealing examples of co-analytic and omega-level sets in the Wadge hierarchy.

Contribution

It introduces new examples of complex sets of infinite sentences, including co-analytic and omega-level sets, within the framework of descriptive set theory.

Findings

Identification of true co-analytic sets in the context of infinite sentences.

Construction of a natural example of a set at the omega level of the Wadge hierarchy.

Analysis of the case where the dictionary is finite, illustrating the set's complexity.

Abstract

We study the sets of the infinite sentences constructible with a dictionary over a finite alphabet, from the viewpoint of descriptive set theory. Among other things, this gives some true co-analytic sets. The case where the dictionary is finite is studied and gives a natural example of a set at the level omega of the Wadge hierarchy.