Generic absoluteness and boolean names for elements of a Polish space

TL;DR

This paper extends a known duality in set theory relating boolean names for complex numbers to a broader class of elements in Polish spaces, and explores its implications with forcing and generic absoluteness.

Contribution

It generalizes Jech's duality to describe boolean names for elements of any Polish space within boolean valued models.

Findings

Describes boolean names for Polish space elements as functions on Stone spaces

Connects duality with generic absoluteness and forcing techniques

Provides a framework for analyzing the theory of these function spaces

Abstract

It is common knowledge in the set theory community that there exists a duality relating the commutative $C^*$-algebras with the family of $B$-names for complex numbers in a boolean valued model for set theory $V^B$. Several aspects of this correlation have been considered in works of the late $1970$'s and early $1980$'s, for example by Takeuti, and by Jech. Generalizing Jech's results, we extend this duality so as to be able to describe the family of boolean names for elements of any given Polish space $Y$ (such as the complex numbers) in a boolean valued model for set theory $V^B$ as a space $C^+(X,Y)$ consisting of functions $f$ whose domain $X$ is the Stone space of $B$, and whose range is contained in $Y$ modulo a meager set. We also outline how this duality can be combined with generic absoluteness results in order to analyze, by means of forcing arguments, the theory of $C^+(X,Y)$.