Quotients, pure existential completions and Arithmetic Universes
Bibliographic record
Abstract
We provide a new description of Joyal's arithmetic universes through a characterization of the exact and regular completions of pure existential completions.We show that the regular and exact completions of the pure existential completion of an elementary doctrine P are equivalent to the reg/lex and ex/lex-completions, respectively, of the category of predicates of P .This result generalizes a previous one by the first author with F. Pasquali and G. Rosolini about doctrines equipped with Hilbert's -operators.Thanks to this characterization, each arithmetic universe in the sense of Joyal can be seen as the exact completion of the pure existential completion of the doctrine of predicates of its Skolem theory.In particular, the initial arithmetic universe in the standard category of ZFC-sets turns out to be the completion with exact quotients of the doctrine of recursively enumerable predicates.
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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 | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| 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".