A generic property of Solovay's set $\Sigma$

TL;DR

This paper demonstrates that Solovay's set  is generic over the ground model using a specific forcing notion that extends the order relation, revealing a fundamental property of .

Contribution

It establishes a new genericity property of Solovay's set  through a novel forcing construction that extends the existing order relation.

Findings

Solovay's set  is shown to be generic over the ground model.

The forcing notion used extends the order relation  with .

This property reveals a fundamental aspect of Solovay's set .

Abstract

We prove that Solovay's set $\Sigma$ is generic over the ground model via a forcing notion whose order relation $\subseteq$-extends the given order relation.