Bibliographic record
Abstract
Our aim in the previous section was to present Gödel's first incompleteness theorem in the context of recursion theory. Much as this “modern” approach is valuable for showing the links between unprovability and uncomputability, it has obscured the simplicity of Gödel's ingenious idea (as it was carried out in his original paper (1931)). What he had accomplished in that paper, through arithmetization of formulas and proofs, was to build a sentence of arithmetic, ℱ, that said “I am not a theorem”. One can easily prove, metamathematically, that such an ℱ is undecidable, if arithmetic is ω-consistent . To see this at the intuitive level, let us replace ω-consistency by correctness. Then surely ℱ is not provable, for if it is, then it is a theorem, and hence false (contradicting correctness). On the other hand, we have just concluded that ℱ is true! Hence, ¬ℱ is false, and therefore not provable either (by correctness). This simple application of the “liar's paradox” is at the heart of the first incompleteness theorem. Imagine now that the arithmetization is actually carried out within (some) formal arithmetic, and that with some effort we have managed to embed into formal arithmetic the metamathematical argument that leads to the assertion “if arithmetic is consistent, then ⊬ ℱ”. The quoted statement is formalized by “Con → ℱ”, where “Con” is some (formal) sentence that says that arithmetic is consistent.
Fetched live from OpenAlex and de-inverted. Abstracts are not stored in this database: the inverted indexes are 8.6 GB of the frame’s 9.3 GB of text, and the host has 13 GB free.
How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.004 | 0.010 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.002 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.002 | 0.005 |
| Scholarly communication | 0.004 | 0.009 |
| Open science | 0.002 | 0.003 |
| Research integrity | 0.001 | 0.005 |
| Insufficient payload (model declined to judge) | 0.009 | 0.002 |
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 source (direct Gemma or distilled Codex), 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".