Comparing theorems of hyperarithmetic analysis with the arithmetic Bolzano-Weierstrass theorem
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Bibliographic record
Abstract
In 1975 H. Friedman introduced two statements of hyperarithmetic analysis, <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 L sans-serif 0"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="sans-serif">S</mml:mi> <mml:msub> <mml:mi mathvariant="sans-serif">L</mml:mi> <mml:mn mathvariant="sans-serif">0</mml:mn> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathsf {SL_0}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> (sequential limit system) and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sans-serif upper A sans-serif upper B sans-serif upper W sans-serif 0"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="sans-serif">A</mml:mi> <mml:mi mathvariant="sans-serif">B</mml:mi> <mml:msub> <mml:mi mathvariant="sans-serif">W</mml:mi> <mml:mn mathvariant="sans-serif">0</mml:mn> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathsf {ABW_0}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> (arithmetic Bolzano-Weierstrass), which are motivated by standard and well-known theorems from analysis such as the Bolzano-Weierstrass theorem for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper F Subscript sigma"> <mml:semantics> <mml:msub> <mml:mi>F</mml:mi> <mml:mi> σ </mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">F_\sigma</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="upper G Subscript delta"> <mml:semantics> <mml:msub> <mml:mi>G</mml:mi> <mml:mi> δ </mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">G_\delta</mml:annotation> </mml:semantics> </mml:math> </inline-formula> sets of reals. In this article we characterize the reverse mathematical strength of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sans-serif upper A sans-serif upper B sans-serif upper W sans-serif 0"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="sans-serif">A</mml:mi> <mml:mi mathvariant="sans-serif">B</mml:mi> <mml:msub> <mml:mi mathvariant="sans-serif">W</mml:mi> <mml:mn mathvariant="sans-serif">0</mml:mn> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathsf {ABW_0}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> by comparing it to most known theories of hyperarithmetic analysis. In particular we show that, 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 sans-serif 0 sans-serif plus sans-serif upper I sans-serif upper Sigma sans-serif 1 Superscript sans-serif 1"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="sans-serif">R</mml:mi> <mml:mi mathvariant="sans-serif">C</mml:mi> <mml:msub> <mml:mi mathvariant="sans-serif">A</mml:mi> <mml:mn mathvariant="sans-serif">0</mml:mn> </mml:msub> <mml:mo mathvariant="sans-serif">+</mml:mo> <mml:mi mathvariant="sans-serif">I</mml:mi> <mml:msubsup> <mml:mi mathvariant="sans-serif"> Σ </mml:mi> <mml:mn mathvariant="sans-serif">1</mml:mn> <mml:mn mathvariant="sans-serif">1</mml:mn> </mml:msubsup> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathsf {RCA_0+I\Sigma ^1_1}</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 L sans-serif 0"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="sans-serif">S</mml:mi> <mml:msub> <mml:mi mathvariant="sans-serif">L</mml:mi> <mml:mn mathvariant="sans-serif">0</mml:mn> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathsf {SL_0}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is equivalent to <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sans-serif upper Sigma sans-serif 1 Superscript sans-serif 1 Baseline minus sans-serif upper A sans-serif upper C sans-serif 0"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msubsup> <mml:mi mathvariant="sans-serif"> Σ </mml:mi> <mml:mn mathvariant="sans-serif">1</mml:mn> <mml:mn mathvariant="sans-serif">1</mml:mn> </mml:msubsup> <mml:mo mathvariant="sans-serif"> − </mml:mo> <mml:mi mathvariant="sans-serif">A</mml:mi> <mml:msub> <mml:mi mathvariant="sans-serif">C</mml:mi> <mml:mn mathvariant="sans-serif">0</mml:mn> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathsf {\Sigma ^1_1-AC_0}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , and that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sans-serif upper A sans-serif upper B sans-serif upper W sans-serif 0"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="sans-serif">A</mml:mi> <mml:mi mathvariant="sans-serif">B</mml:mi> <mml:msub> <mml:mi mathvariant="sans-serif">W</mml:mi> <mml:mn mathvariant="sans-serif">0</mml:mn> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathsf {ABW_0}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is implied by <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sans-serif upper Sigma s
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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 | 0.001 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.000 | 0.003 |
| Science and technology studies | 0.000 | 0.002 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.002 | 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