Beth definability and the Stone-Weierstrass Theorem

TL;DR

This paper links the classical Stone-Weierstrass Theorem to Beth definability in an equational logic framework, establishing a logical foundation for a fundamental theorem in functional analysis.

Contribution

It introduces an infinitary equational logic and shows the Stone-Weierstrass Theorem follows from Beth definability within this logical setting.

Findings

Established a logical interpretation of the Stone-Weierstrass Theorem.

Proved a strong completeness theorem for the associated propositional logic.

Connected functional analysis results with logical definability properties.

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