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arXiv 2007.052812020-07math.LOmath.FAmath.GNmath.RA

Beth definability and the Stone-Weierstrass Theorem

Luca Reggio

备注：: 27 pages. v3: presentation improved throughout; added a new Section 5 establishing a strong completeness result

英文摘要：

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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