# Definable ${\mathsf E}_0$ classes at arbitrary projective levels

## TL;DR

This paper constructs a model of ZFC where, for any given projective level n≥3, there exists a non-OD E_0 equivalence class that is a countable, lightface Pi^1_n set, contrasting with the OD nature of non-empty Sigma^1_n sets.

## Contribution

It introduces a modified invariant Jensen forcing to create models with specific definability properties of E_0 classes at arbitrary projective levels.

## Key findings

- Existence of non-OD E_0 classes at any projective level n≥3.
- Non-empty Sigma^1_n sets are necessarily constructible and OD.
- The model distinguishes between E_0 classes and Sigma^1_n sets in terms of OD properties.

## Abstract

Using a modification of the invariant Jensen forcing, we define a model of ZFC, in which, for a given $n\ge3$, there exists a lightface $\varPi^1_n$ set of reals, which is a ${\mathsf E}_0$ equivalence class, hence a countable set, and which does not contain any OD element, while every non-empty countable $\varSigma^1_n$ set of reals is necessarily constructible, hence contains only OD reals.