Foundation

Formalizing mathematical logic in Lean 4.

Structure & Summary

Main results of this repository. More detailed explanations are provided in the Book and Doc.

• Propositional: Propositional Logic
  • Tait-style Calculus and Completeness
  • Completeness for Kripke semantics
  • Disjunctive Property of intuitionistic logic
  • Propositional Logic Zoo
• FirstOrder: First-Order Logic
  • Completeness Theorem
  • Cut-Elimination of First-Order Sequent Calculus (Gentzen's Hauptsatz)
  • Gödel-Gentzen Translation
  • Downward Löwenheim-Skolem Theorem
  • Arithmetic
    • Gödel's First and Second Incompleteness Theorems
    • Arithmetic Theory Zoo
  • Set Theory
    • Consistency of ZFC
    • Set Theory Zoo
• Modal: Basic Modal Logic (with modal operators )
  • Kripke Semantics
  • Neighborhood Semantics
  • Gödel-McKinsey-Tarski Theorem and Modal Companions
  • Boxdot Conjecture
  • Modal Logic Zoo
• ProvabilityLogic: Provability Logic
  • Solovay's Arithmetical Completeness Theorem for GL
• InterpretabilityLogic: Interpretability Logic
  • Veltman Semantics
  • Verbugge Semantics (a.k.a generalized Veltman semantics)
  • Interpretability Logic Zoo
• Meta: Proof automation.
• Logic: Fundamental tools for various logics.
• Vospiel: Supplemental definitions and theorems for mathlib.

Documents

• Book, summary of results, generated by verso.
• Doc, catelogue of definitions and theorems, generated by doc-gen4.

Zoo

Automatically generated diagrams "Zoo", illustrate the Lean4-verified interrelationships among proof systems.

• A solid arrow indicates that is strictly stronger than ; that is, is stronger than , while is not stronger than , in terms of provability strength.
• A dashed arrow indicates that is stronger than in terms of provability strength.
• A double line indicates that and are equivalent in terms of provability strength.

Arithmetic Theory Zoo

PDF version

Set Theory Zoo

PDF version

Propositional Logic Zoo

PDF version

Modal Logic Zoo

PDF version

Interpretability Logic Zoo

PDF version

Build

Book

Book is automatically generated by verso.

Follow the instruction in Book/README.md and build. Some tools e.g. miniserve or python3 -m http.server are useful for check locally.

lake exe book

miniserve _out/html-multi --index index.html

Developers

List of contact information and areas of expertise of the current main developers. If you have any interest or questions, create a new issue or contact us directly.

• Palalansoukî (Shogo Saito, @iehality, ✉️:palalansouki@gmail.com)
  • Overall design and maintenance.
  • First-order logic.
  • Intuitionistic first-order logic.
  • Proof automation.
  • Provability logic.
• SnO2WMaN (Mashu Noguchi, @SnO2WMaN, ✉️:me@sno2wman.net)
  • Modal logic.
  • Propositional logic (including intermediate logic).
  • Provability logic.
  • Interpretability Logic.
  • Miscellaneous repository maintenance (e.g. GitHub Actions)

If you with to cite this repository in academic papers, refer to CITATION.cff.

Financial Supports

Any financial supports would be grateful. If you considered, please contact us.

Previous Sponsors

Companies and organizations that have supported us in the past.

• Proxima Technology (2024-2025)