Infinitary logic and basically disconnected compact Hausdorff spaces
2018
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 $[0,1]$.
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