On the role of the collection principle for Σ⁰₂-formulas in second-order reverse mathematics
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Bibliographic record
Abstract
We show that the principle <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sans-serif upper P sans-serif upper A sans-serif upper R sans-serif upper T"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="sans-serif">P</mml:mi> <mml:mi mathvariant="sans-serif">A</mml:mi> <mml:mi mathvariant="sans-serif">R</mml:mi> <mml:mi mathvariant="sans-serif">T</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathsf {PART}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> from Hirschfeldt and Shore is equivalent to the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Sigma 2 Superscript 0"> <mml:semantics> <mml:msubsup> <mml:mi mathvariant="normal"> Σ </mml:mi> <mml:mn>2</mml:mn> <mml:mn>0</mml:mn> </mml:msubsup> <mml:annotation encoding="application/x-tex">\Sigma ^0_2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -Bounding principle <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper B normal upper Sigma 2 Superscript 0"> <mml:semantics> <mml:mrow> <mml:mi>B</mml:mi> <mml:msubsup> <mml:mi mathvariant="normal"> Σ </mml:mi> <mml:mn>2</mml:mn> <mml:mn>0</mml:mn> </mml:msubsup> </mml:mrow> <mml:annotation encoding="application/x-tex">B\Sigma ^0_2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> over <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sans-serif upper R sans-serif upper C sans-serif upper A Subscript 0"> <mml:semantics> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="sans-serif">R</mml:mi> <mml:mi mathvariant="sans-serif">C</mml:mi> <mml:mi mathvariant="sans-serif">A</mml:mi> </mml:mrow> <mml:mn>0</mml:mn> </mml:msub> <mml:annotation encoding="application/x-tex">\mathsf {RCA}_0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , answering one of their open questions. Furthermore, we also fill a gap in a proof of Cholak, Jockusch and Slaman by showing that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper D 2 squared"> <mml:semantics> <mml:msubsup> <mml:mi>D</mml:mi> <mml:mn>2</mml:mn> <mml:mn>2</mml:mn> </mml:msubsup> <mml:annotation encoding="application/x-tex">D^2_2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> implies <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper B normal upper Sigma 2 Superscript 0"> <mml:semantics> <mml:mrow> <mml:mi>B</mml:mi> <mml:msubsup> <mml:mi mathvariant="normal"> Σ </mml:mi> <mml:mn>2</mml:mn> <mml:mn>0</mml:mn> </mml:msubsup> </mml:mrow> <mml:annotation encoding="application/x-tex">B\Sigma ^0_2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and is thus indeed equivalent to Stable Ramsey’s Theorem for Pairs ( <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sans-serif upper S sans-serif upper R sans-serif upper T Subscript 2 Superscript 2"> <mml:semantics> <mml:msubsup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="sans-serif">S</mml:mi> <mml:mi mathvariant="sans-serif">R</mml:mi> <mml:mi mathvariant="sans-serif">T</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> <mml:mn>2</mml:mn> </mml:msubsup> <mml:annotation encoding="application/x-tex">\mathsf {SRT}^2_2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> ). This also allows us to conclude that the combinatorial principles <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sans-serif upper I sans-serif upper P sans-serif upper T Subscript 2 Superscript 2"> <mml:semantics> <mml:msubsup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="sans-serif">I</mml:mi> <mml:mi mathvariant="sans-serif">P</mml:mi> <mml:mi mathvariant="sans-serif">T</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> <mml:mn>2</mml:mn> </mml:msubsup> <mml:annotation encoding="application/x-tex">\mathsf {IPT}^2_2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sans-serif upper S sans-serif upper P sans-serif upper T Subscript 2 Superscript 2"> <mml:semantics> <mml:msubsup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="sans-serif">S</mml:mi> <mml:mi mathvariant="sans-serif">P</mml:mi> <mml:mi mathvariant="sans-serif">T</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> <mml:mn>2</mml:mn> </mml:msubsup> <mml:annotation encoding="application/x-tex">\mathsf {SPT}^2_2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sans-serif upper S sans-serif upper I sans-serif upper P sans-serif upper T Subscript 2 Superscript 2"> <mml:semantics> <mml:msubsup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="sans-serif">S</mml:mi> <mml:mi mathvariant="sans-serif">I</mml:mi> <mml:mi mathvariant="sans-serif">P</mml:mi> <mml:mi mathvariant="sans-serif">T</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> <mml:mn>2</mml:mn> </mml:msubsup> <mml:annotation encoding="application/x-tex">\mathsf {SIPT}^2_2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> defined by Dzhafarov and Hirst all imply <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathM
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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 | 0.002 | 0.002 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.000 | 0.002 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.003 | 0.001 |
| Research integrity | 0.000 | 0.001 |
| 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