# A symmetric $\beta$-model

## TL;DR

This paper constructs a countable $eta$-model where definability between reals coincides with hyperarithmetical reducibility, providing insights into the structure of models in descriptive set theory.

## Contribution

It proves the existence of a countable $eta$-model with a unique definability property linking reals and hyperarithmetical reducibility, advancing understanding of $eta$-models.

## Key findings

- Existence of a countable $eta$-model with specific definability properties
- Equivalence of definability and hyperarithmetical reducibility within the model
- Related results and open questions in the theory of $eta$-models

## Abstract

We prove that there exists a countable $\beta$-model in which, for all reals $X$ and $Y$, $X$ is definable from $Y$ if and only $X$ is hyperarithmetical in $Y$. We also obtain some related results and pose some related questions.