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<p>The article offers a novel reconstruction of Hilbert’s early metatheory of formal axiomatics. His foundational work from the turn of the last century is often regarded as a central contribution to a “model-theoretic” viewpoint in modern logic and mathematics. The article will re-assess Hilbert’s role in the development of model theory by focusing on two aspects of his contributions to the axiomatic foundations of geometry and analysis. First, we examine Hilbert’s conception of mathematical theories and their interpretations; in particular, we argue that his early semantic views can be understood in terms of a notion of <i>translational isomorphism</i> between models of an axiomatic theory. Second, we offer a logical reconstruction of his consistency and independence results in geometry in terms of the notion of <i>interpretability</i> between theories.</p>

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Hilbert’s Early Metatheory Revisited

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2025年5月13日

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  • Eduardo N. Giovannini,
  • Georg Schiemer

摘要

The article offers a novel reconstruction of Hilbert’s early metatheory of formal axiomatics. His foundational work from the turn of the last century is often regarded as a central contribution to a “model-theoretic” viewpoint in modern logic and mathematics. The article will re-assess Hilbert’s role in the development of model theory by focusing on two aspects of his contributions to the axiomatic foundations of geometry and analysis. First, we examine Hilbert’s conception of mathematical theories and their interpretations; in particular, we argue that his early semantic views can be understood in terms of a notion of translational isomorphism between models of an axiomatic theory. Second, we offer a logical reconstruction of his consistency and independence results in geometry in terms of the notion of interpretability between theories.