Bibliographic record
Abstract
The replacement (or collection or choice) axiom scheme BB(Γ) asserts bounded quantifier exchange as follows: ∀i < |a| ∃x < aϕ(i,x) → ∃w ∀i < |a|ϕ(i,[w]i), for ϕ in the class Γ of formulas. The theory S12 proves the scheme BB(Σb1), and thus in S12 every Σb1 formula is equivalent to a strict Σb1 formula (in which all non-sharply-bounded quantifiers are in front). Here we prove (sometimes subject to an assumption) that certain theories weaker than S12 do not prove either BB(Σb1) or BB(Σb0). We show (unconditionally) that V 0 does not prove BB(Σb0), where V0 (essentially IΣ1,b0) is the two-sorted theory associated with the complexity class AC0. We show that PV does not prove BB(Σb0), assuming that integer factoring is not possible in probabilistic polynomial time. Johannsen and Pollett introduced the theory C02 associated with the complexity class TC0, and later introduced an apparently weaker theory Δb1 − CR for the same class. We use our methods to show that Δb1 − CR is indeed weaker than C02, assuming that RSA is secure against probabilistic polynomial time attack.Our main tool is the KPT witnessing theorem.
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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 | 0.007 | 0.015 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.002 |
| Bibliometrics | 0.002 | 0.001 |
| Science and technology studies | 0.003 | 0.013 |
| Scholarly communication | 0.006 | 0.014 |
| Open science | 0.003 | 0.009 |
| Research integrity | 0.002 | 0.006 |
| 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".