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Hilbert's Early Metatheory Revisited.

Eduardo N Giovannini1, Georg Schiemer2

  • 1CONICET/Universidad Nacional del Litoral, Paraje El Pozo, 3000 Santa Fe, Argentina.

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|March 16, 2026
PubMed
Summary

This study reconstructs David Hilbert's early formal axiomatic metatheory, highlighting his contributions to model theory and the semantic views of mathematical theories.

Area of Science:

  • Mathematical Logic
  • Foundations of Mathematics
  • History of Mathematics

Background:

  • David Hilbert's early work on formal axiomatics is a cornerstone of modern logic and mathematics.
  • His contributions are often viewed through a "model-theoretic" lens.
  • Re-evaluating his foundational role in model theory is crucial for understanding the development of mathematical thought.

Purpose of the Study:

  • To offer a novel reconstruction of Hilbert's early metatheory of formal axiomatics.
  • To re-assess Hilbert's role in the development of model theory.
  • To examine his contributions to the axiomatic foundations of geometry and analysis.

Main Methods:

  • Focusing on Hilbert's conception of mathematical theories and their interpretations.

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  • Analyzing his early semantic views through "translational isomorphism" between models.
  • Logically reconstructing his consistency and independence results using "interpretability" between theories.

    • Main Results:

    • Hilbert's early semantic views can be understood via "translational isomorphism" of models.
    • His consistency and independence results in geometry are reconstructible through "interpretability" of theories.
    • This provides a new perspective on Hilbert's foundational contributions.

    Conclusions:

    • Hilbert's early work laid crucial groundwork for model theory.
    • The concepts of "translational isomorphism" and "interpretability" offer novel insights into his metatheory.
    • This reconstruction deepens our understanding of the historical development of formal axiomatic systems.