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A Succinct Formalization of the Completeness of First-Order Logic

Author Asta Halkjær From

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Asta Halkjær From
  • Technical University of Denmark, Kongens Lyngby, Denmark


Thanks to Frederik Krogsdal Jacobsen, Alexander Birch Jensen and Jørgen Villadsen for their useful comments. I am especially grateful to the anonymous reviewers for their insightful remarks which have improved both the formalization and the paper.

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Asta Halkjær From. A Succinct Formalization of the Completeness of First-Order Logic. In 27th International Conference on Types for Proofs and Programs (TYPES 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 239, pp. 8:1-8:24, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)


I succinctly formalize the soundness and completeness of a small Hilbert system for first-order logic in the proof assistant Isabelle/HOL. The proof combines and details ideas from de Bruijn, Henkin, Herbrand, Hilbert, Hintikka, Lindenbaum, Smullyan and others in a novel way, and I use a declarative style, custom notation and proof automation to obtain a readable formalization. The formalized definitions of Hintikka sets and Herbrand structures allow open and closed formulas to be treated uniformly, making free variables a non-concern. This paper collects important techniques in mathematical logic in a way suited for both study and further work.

Subject Classification

ACM Subject Classification
  • Theory of computation → Logic and verification
  • First-Order Logic
  • Completeness
  • Isabelle/HOL


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