add(Incompleteness): First incompleteness theorem from the halting problem#836
Merged
Conversation
Co-Authored-By: Claude Opus 4.8 <noreply@anthropic.com>
Co-Authored-By: Claude Opus 4.8 <noreply@anthropic.com>
This file contains hidden or bidirectional Unicode text that may be interpreted or compiled differently than what appears below. To review, open the file in an editor that reveals hidden Unicode characters.
Learn more about bidirectional Unicode characters
Sign up for free
to join this conversation on GitHub.
Already have an account?
Sign in to comment
Add this suggestion to a batch that can be applied as a single commit.This suggestion is invalid because no changes were made to the code.Suggestions cannot be applied while the pull request is closed.Suggestions cannot be applied while viewing a subset of changes.Only one suggestion per line can be applied in a batch.Add this suggestion to a batch that can be applied as a single commit.Applying suggestions on deleted lines is not supported.You must change the existing code in this line in order to create a valid suggestion.Outdated suggestions cannot be applied.This suggestion has been applied or marked resolved.Suggestions cannot be applied from pending reviews.Suggestions cannot be applied on multi-line comments.Suggestions cannot be applied while the pull request is queued to merge.Suggestion cannot be applied right now. Please check back later.
Derives Gödel's first incompleteness theorem from the undecidability of the halting problem, completing the work started (and abandoned) in #508.
Changes
Foundation/FirstOrder/Incompleteness/Halting.lean:REPred.iff_decoded_pred/ComputablePred.iff_decoded_pred: for aPrimcodabletypeα, r.e.-ness / computability of a predicateA : α → Propis equivalent to that of the correspondingℕ-predicate obtained viadecode. Pure mathlib computability lemmas (composition withdecode/encode).incomplete_of_REPred_not_ComputablePred:Primcodablegeneralization of the existingincomplete_of_REPred_not_ComputablePred_Nat. Needed because the halting predicate lives overNat.Partrec.Code, notℕ.incomplete_of_halting_problem: the halting problem is r.e. but not recursive (ComputablePred.halting_problem_re/ComputablePred.halting_problem), hence incompleteness.The core diagonalization lemma
incomplete_of_REPred_not_ComputablePred_Natalready exists onmaster.Verification
lake build Foundation.FirstOrder.Incompleteness.Haltingsucceeds (forced rebuild), no errors/warnings/sorry.incomplete_of_halting_problemdepends only onpropext,Classical.choice,Quot.sound(nosorryAx, no custom axioms).Supersedes #508.
🤖 Generated with Claude Code