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Explore Beth's definability theorem and its applications in model theory through this 59-minute mathematical lecture. Discover how this fundamental theorem establishes a powerful criterion connecting implicit and explicit definability, demonstrating that if a predicate P in an expanded language L′ is preserved across all models of an L′-theory that share the same reduct to a smaller language L, then P must be definable by a formula in L. Learn how this principle serves as a general method for understanding definability across various mathematical contexts, with particular emphasis on its remarkable compatibility with the model theory of henselian valued fields and ordered abelian groups. Examine natural characterizations of fields and groups with automatic definability, specifically those where every henselian valuation with prescribed residue field or value group becomes definable in the pure language of rings, whether with or without parameters. Gain insight into how this framework provides a unified perspective on recent definability results by connecting them to intrinsic properties of residue fields and value groups, based on collaborative research with B. Boissonneau, F. Jahnke, and P. Touchard.
