Borel functors, interpretations, and strong conceptual completeness for ℒ_{𝜔₁𝜔}
Bibliographic record
Abstract
We prove a strong conceptual completeness theorem (in the sense of Makkai) for the infinitary logic L ω 1 ω \mathcal {L}_{\omega _1\omega } : every countable L ω 1 ω \mathcal {L}_{\omega _1\omega } -theory can be canonically recovered from its standard Borel groupoid of countable models, up to a suitable syntactical notion of equivalence. This implies that given two theories ( L , T ) (\mathcal {L}, \mathcal {T}) and ( L ′ , T ′ ) (\mathcal {L}’, \mathcal {T}’) (in possibly different languages L , L ′ \mathcal {L}, \mathcal {L}’ ), every Borel functor Mod ( L ′ , T ′ ) → Mod ( L , T ) \text {Mod}(\mathcal {L}’, \mathcal {T}’) \to \text {Mod}(\mathcal {L}, \mathcal {T}) between the respective groupoids of countable models is Borel naturally isomorphic to the functor induced by some L ω
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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 | 0.006 | 0.008 |
| Meta-epidemiology (narrow) | 0.001 | 0.002 |
| Meta-epidemiology (broad) | 0.001 | 0.005 |
| Bibliometrics | 0.003 | 0.002 |
| Science and technology studies | 0.005 | 0.009 |
| Scholarly communication | 0.006 | 0.017 |
| Open science | 0.002 | 0.009 |
| Research integrity | 0.002 | 0.007 |
| Insufficient payload (model declined to judge) | 0.010 | 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".