Enumerations of families closed under finite differences
Bibliographic record
Abstract
Slaman and Wehner independently built a family of sets with the property that every non-computable degree can compute an enumeration of the family, but there is no computable enumeration of the family. We call such a family a Slaman–Wehner family. The original Slaman–Wehner argument relies on all sets in the family constructed being finite, and in particular, it diagonalizes against computably enumerated families using only finite differences. In this paper we ask whether this is a necessary feature, that is, whether there is a Slaman–Wehner family closed under finite differences. This question remains open but we obtain a number of interesting partial results which can be interpreted as saying that the question is quite hard. First of all, no Slaman–Wehner family closed under finite differences can contain a finite set, and the enumeration of the family from a non-computable degree cannot be uniform (whereas, in the Slaman–Wehner construction, it is uniform). On the other hand, we build the following examples of families closed under finite differences which show the impossibility of several natural attempts to show that no Slaman–Wehner family exists: (1) a family that can be enumerated by every non-low degree, but not by any low degree; (2) a family that can be enumerated by any set in a given uniform list of c.e. sets, but which cannot be enumerated computably; and (3) a family that can be enumerated by a given [Formula: see text] set, but which cannot be computably enumerated.
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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 | 0.003 | 0.013 |
| Meta-epidemiology (narrow) | 0.000 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.002 |
| Bibliometrics | 0.002 | 0.001 |
| Science and technology studies | 0.002 | 0.003 |
| Scholarly communication | 0.002 | 0.008 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.005 | 0.001 |
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".