Beth definability and the Stone-Weierstrass Theorem

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 ${\models }_{\Delta }$ associated with an infinitary variety $\Delta$ and show that the Stone-Weierstrass Theorem is a consequence of the Beth definability property of ${\models }_{\Delta }$, stating that every implicit definition can be made explicit. Further, we define an infinitary propositional logic ${⊢}_{\Delta }$ by means of a Hilbert-style calculus and prove a strong completeness result whereby the semantic notion of consequence associated with ${⊢}_{\Delta }$ coincides with ${\models }_{\Delta }$.

arXiv comment27 pages. v3: presentation improved throughout; added a new Section 5 establishing a strong completeness result