https://github.com/FormalizedFormalLogic/Foundation
Lean4 Logic Formalization
https://github.com/FormalizedFormalLogic/Foundation
incompleteness lean4 logic
Last synced: over 1 year ago
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Lean4 Logic Formalization
- Host: GitHub
- URL: https://github.com/FormalizedFormalLogic/Foundation
- Owner: FormalizedFormalLogic
- License: apache-2.0
- Created: 2023-02-11T18:48:33.000Z (over 3 years ago)
- Default Branch: master
- Last Pushed: 2024-07-25T21:55:05.000Z (almost 2 years ago)
- Last Synced: 2024-07-25T23:28:12.305Z (almost 2 years ago)
- Topics: incompleteness, lean4, logic
- Language: Lean
- Homepage: https://iehality.github.io/lean4-logic/book/
- Size: 2.1 MB
- Stars: 49
- Watchers: 5
- Forks: 4
- Open Issues: 13
-
Metadata Files:
- Readme: README.md
- License: LICENSE
Awesome Lists containing this project
README
[Book]: https://formalizedformallogic.github.io/Foundation/book
[Doc]: https://FormalizedFormalLogic.github.io/Foundation/doc
# Foundation
Formalizing mathematical logics in Lean 4.
## Summary
Main Result in this repository. More results and details are in [Book] and [Doc].
- [Propositional Logic][prop]
- [Tait-style calculus][prop:classical_tait] and [completeness][prop:classical_tait_complete]
- [Completeness for Kripke semantics][prop:kripke]
- [Disjunctive Property of intuitionistic logic][prop:int_disjunctive]
- Rejection Law of Excluded Middle in intuitionistic logic and [sublogic relations][prop:sublogics]
- [First-Order Logic][first_order] and [Arithmetics][arith]
- [Completeness Theorem][first_order:completeness]
- [Gödel-Gentzen Translation][first_order:goedel_translation]
- [Cut-elimination of first-order sequent calculus _(Gentzen's Hauptsatz)_][first_order:haupstaz]
- [Arithmetic][arith] and [Arithmetization](arithmetization)
- Gödel's [First][arith:goedel_it1] and [Second][arith:goedel_it2] Incompleteness Theorems
- [Basic Modal Logic][modal:logic] (with modal operators $\Box, \Diamond$)
- [Kripke completeness for well-known subsystems][modal:logic_kripke_completeness]
- [_Modal Cube_][modal:cube], and [sublogic relations for other logics](modal:sublogic)
- [Gödel-McKinsey-Tarski Theorem][modal:gmt_theorem] and [Modal Companions](modal:companion)
- [Provability Logic][provability_logic]
[prop]: ./Foundation/Propositional
[prop:classical_tait]: ./Foundation/Propositional/Tait/Calculus.lean
[prop:classical_tait_complete]: ./Foundation/Propositional/Classical/Tait.lean
[prop:classical_complete]: ./Foundation/Propositional/Classical/Tait.lean
[prop:kripke]: ./Foundation/Propositional/Kripke
[prop:int_disjunctive]: ./Foundation/Propositional/Kripke/Hilbert/Int.lean
[prop:sublogics]: ./Foundation/Propositional/Logic/Sublogic.lean
[first_order]: https://formalizedformallogic.github.io/Foundation/book/first_order/index.html
[first_order:completeness]: https://formalizedformallogic.github.io/Foundation/book/first_order/completeness.html
[first_order:haupstaz]: ./Foundation/FirstOrder/Hauptsatz.lean
[first_order:goedel_translation]: ./Foundation/IntFO/Translation.lean
[arith]: https://formalizedformallogic.github.io/Foundation/book/first_order/arithmetics.html
[arithmetization]: ./Foundation/Arithmetization
[arith:goedel_it1]: https://formalizedformallogic.github.io/Foundation/book/first_order/goedel1.html
[arith:goedel_it2]: https://formalizedformallogic.github.io/Foundation/book/first_order/goedel2.html
[modal:logic]: ./Foundation/Modal
[modal:logic_kripke_completeness]: ./Foundation/Modal/Kripke/Hilbert
[modal:cube]: ./Foundation/Modal/Logic/Sublogic/ModalCube.lean
[modal:sublogic]: ./Foundation/Modal/Logic/Sublogic
[modal:gmt_theorem]: ./Foundation/Modal/ModalCompanion/Int.lean
[modal:companion]: ./Foundation/Modal/ModalCompanion
[provability_logic]: ./Foundation/Incompleteness/ProvabilityLogic
## Documents
- [Book], summary of results.
- [Doc], documentation generated by [doc-gen4](https://github.com/leanprover/doc-gen4).
## Sponsor
This project is supported by [Proxima Technology].
[
][Proxima Technology]
[Proxima Technology]: https://proxima-ai-tech.com/