Beth definability and the Stone-Weierstrass Theorem

Luca Reggio

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arXiv · 2007.05281

Beth definability and the Stone-Weierstrass Theorem

Luca Reggio

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Abstract

The Stone-Weierstrass Theorem for compact Hausdorff spaces is a basic result of functional analysis with far-reaching consequences. We introduce an equational logic $\vDash_Δ$ associated with an infinitary variety $Δ$ and show that the Stone-Weierstrass Theorem is a consequence of the Beth definability property of $\vDash_Δ$, stating that every implicit definition can be made explicit. Further, we define an infinitary propositional logic $\vdash_Δ$ by means of a Hilbert-style calculus and prove a strong completeness result whereby the semantic notion of consequence associated with $\vdash_Δ$ coincides with $\vDash_Δ$.

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BibTeX RIS

Luca Reggio. 2021-05-06. Beth definability and the Stone-Weierstrass Theorem. https://arxiv.org/abs/2007.05281

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