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 $\vDash_{\Delta}$ 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 $\vDash_{\Delta}$.
- Publication:
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arXiv e-prints
- Pub Date:
- July 2020
- DOI:
- 10.48550/arXiv.2007.05281
- arXiv:
- arXiv:2007.05281
- Bibcode:
- 2020arXiv200705281R
- Keywords:
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- Mathematics - Logic;
- Mathematics - Functional Analysis;
- Mathematics - General Topology;
- Mathematics - Rings and Algebras
- E-Print:
- 27 pages. v3: presentation improved throughout