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 < a ϕ( i , x ) → ∃ w ∀ i < | a |ϕ( i ,[ w ] i ), for ϕ in the class Γ of formulas. The theory S 1 2 proves the scheme BB(Σ b 1 ), and thus in S 1 2 every Σ b 1 formula is equivalent to a strict Σ b 1 formula (in which all non-sharply-bounded quantifiers are in front). Here we prove (sometimes subject to an assumption) that certain theories weaker than S 1 2 do not prove either BB(Σ b 1 ) or BB(Σ b 0 ). We show (unconditionally) that V 0 does not prove BB(Σ b 0 ), where V 0 (essentially IΣ 1, b 0 ) is the two-sorted theory associated with the complexity class AC 0 . We show that PV does not prove BB(Σ b 0 ), assuming that integer factoring is not possible in probabilistic polynomial time. Johannsen and Pollett introduced the theory C 0 2 associated with the complexity class TC 0 , and later introduced an apparently weaker theory Δ b 1 − CR for the same class. We use our methods to show that Δ b 1 − CR is indeed weaker than C 0 2 , 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 | 0.001 |
| 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