Consistency of the Predicative Calculus of Cumulative Inductive Constructions (pCuIC)
Bibliographic record
Abstract
Gödel published his groundbreaking first incompleteness theorem in 1931, stating that a large class of formal logics admits independent sentences which are neither provable nor refutable. This result, in conjunction with his second incompleteness theorem, established the impossibility of concluding Hilbert’s program, which pursued a possible path towards a single formal system unifying all of mathematics. Using a technical trick to refine Gödel’s original proof, the incompleteness result was strengthened further by Rosser in 1936 regarding the conditions imposed on the formal systems. Computability theory, which also originated in the 1930s, was quickly applied to formal logics by Turing, Kleene, and others to yield incompleteness results similar in strength to Gödel’s original theorem, but weaker than Rosser’s refinement. Only much later, Kleene found an improved but far less well-known proof based on computational notions, yielding a result as strong as Rosser’s. In this expository paper, we work in constructive type theory to reformulate Kleene’s incompleteness results abstractly in the setting of synthetic computability theory and assuming a form of Church’s thesis, an axiom internalising the fact that all functions definable in such a setting are computable. Our novel, greatly condensed reformulation showcases the simplicity of the computational argument while staying formally entirely precise, a combination hard to achieve in typical textbook presentations. As an application, we instantiate the abstract result to first-order logic in order to derive essential incompleteness and, along the way, essential undecidability of Robinson arithmetic. This paper is accompanied by a Coq mechanisation covering all our results and based on existing libraries of undecidability proofs and first-order logic, complementing the extensive work on mechanised incompleteness using the Gödel-Rosser approach. In contrast to the related mechanisations, our development follows Kleene’s ideas and utilises Church’s thesis for additional simplicity.
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How this classification was reachedexpand
Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.003 | 0.002 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one teacher head, not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".