Infinitary logic and basically disconnected compact Hausdorff spaces

TL;DR

This paper introduces an infinitary extension of ukasiewicz logic, characterizing models as algebras over basically disconnected compact Hausdorff spaces, and proves standard completeness with respect to the real interval.

Contribution

It extends ukasiewicz logic to an infinitary system with models linked to basically disconnected spaces and establishes completeness results.

Findings

Models are algebras of Borel functions on basically disconnected spaces.

The logic's Lindenbaum-Tarski algebra corresponds to ukasiewicz Borel functions.

The system is complete with respect to the real interval [0,1].

Abstract

We extend \L ukasiewicz logic obtaining the infinitary logic $\mathcal{IR}\L$ whose models are algebras $C(X,[0,1])$, where $X$ is a basically disconnected compact Hausdorff space. Equivalently, our models are unit intervals in $\sigma$-complete Riesz spaces with strong unit. The Lindenbaum-Tarski algebra of $\mathcal{IR}\L$ is, up to isomorphism, an algebra of $[0,1]$-valued Borel functions. Finally, our system enjoys standard completeness with respect to the real interval $[0,1]$.