Infinitary logic and basically disconnected compact Hausdorff spaces

Abstract

We extend Łukasiewicz logic obtaining the infinitary logic $\mathcal{I}\mathcal{R}Ł$ 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 $\mathcal{I}\mathcal{R}Ł$ is, up to isomorphism, an algebra of $[0,1]$-valued Borel functions. Finally, our system enjoys standard completeness with respect to the real interval $[0,1]$.