The strength of replacement in weak arithmetic
Why this work is in the frame
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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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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 | 0.001 | 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 it