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Gödel's Theorem Without Tears

Essential Incompleteness in Synthetic Computability

This repository contains the Coq code accompanying Dominik Kirst's and Benjamin Peters' CSL 2023 submission (here) as part of a fork of the Coq library of undecidability proofs.

The documentation can found here.

The development can be found at theories/FOL/Incompleteness/ and comprises the following files:

  • utils.v: Utilities for vectors and a definition of partial functions
  • epf.v: Definition of Church's thesis and undecidability/uncomputability results
  • dprm.v: Church's thesis for mu-recursive functions
    • recalg.v: Enumerability and discreteness of mu-recursive functions (by Johannes Hostert)
  • formal_systems.v: Abstract formal systems
  • abstract_incompleteness.v: Incompleteness of abstract formal systems
  • qdec.v: Q-decidability
  • sigma1.v: Σ1-completeness
  • ctq.v: Church's thesis for Q and related results
  • fol_incompleteness.v: Essential incompleteness of Robinson arithmetic

Installation Instructions

If you can use opam 2 on your system, you can follow the instructions here.

Manual installation

You need Coq 8.15 built on OCAML >= 4.07.1, the Smpl package, the Equations package, and the MetaCoq package for Coq. If you are using opam 2 you can use the following commands to install the dependencies on a new switch:

opam switch create coq-library-undecidability 4.07.1+flambda
eval $(opam env)
opam repo add coq-released https://coq.inria.fr/opam/released
opam update
opam pin add -n -y -k git coq-metacoq-template.dev+8.15 "https://github.com/MetaCoq/metacoq.git#9493bb6"
opam install . --deps-only

Building the incompleteness results

  • make all builds the whole library and might take >30min
  • make FOL/Incompleteness/fol_incompleteness.vo compiles all files necessary to assess the incompleteness results in <15min
  • make html generates clickable coqdoc .html in the website subdirectory
  • make clean removes all build files in theories and .html files in the website directory

Compiled interfaces

The library is compatible with Coq's compiled interfaces (vos) for quick infrastructural access.

  • make vos builds compiled interfaces for the library
  • make FOL/Incompleteness/fol_incompleteness.vos builds compiled interfaces for the incompleteness results
  • make vok checks correctness of the library

Troubleshooting

Windows

If you use Visual Studio Code on Windows 10 with Windows Subsystem for Linux (WSL), then local opam switches may cause issues. To avoid this, you can use a non-local opam switch, i.e. opam switch create 4.07.1+flambda.

Coq version

Be careful that this branch only compiles under Coq 8.15. If you want to use a different Coq version you have to change to a different branch.