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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.
Erkenntnis
|March 16, 2026
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.
- 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.
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