Bibliographic record
Abstract
There is a well-known global equivalence between <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Sigma 2 Superscript 1"> <mml:semantics> <mml:msubsup> <mml:mi mathvariant="normal"> Σ </mml:mi> <mml:mn>2</mml:mn> <mml:mn>1</mml:mn> </mml:msubsup> <mml:annotation encoding="application/x-tex">\Sigma ^1_2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> sets having the universal Baire property, two-step <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Sigma 3 Superscript 1"> <mml:semantics> <mml:msubsup> <mml:mi mathvariant="normal"> Σ </mml:mi> <mml:mn>3</mml:mn> <mml:mn>1</mml:mn> </mml:msubsup> <mml:annotation encoding="application/x-tex">\Sigma ^1_3</mml:annotation> </mml:semantics> </mml:math> </inline-formula> generic absoluteness, and the closure of the universe under the sharp operation. In this note, we determine the exact consistency strength of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Sigma 2 Superscript 1"> <mml:semantics> <mml:msubsup> <mml:mi mathvariant="normal"> Σ </mml:mi> <mml:mn>2</mml:mn> <mml:mn>1</mml:mn> </mml:msubsup> <mml:annotation encoding="application/x-tex">\Sigma ^1_2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> sets being <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis 2 Superscript omega Baseline right-parenthesis Superscript plus"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:msup> <mml:mn>2</mml:mn> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi> ω </mml:mi> </mml:mrow> </mml:msup> <mml:msup> <mml:mo stretchy="false">)</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>+</mml:mo> </mml:mrow> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">(2^{\omega })^{+}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -cc-universally Baire, which is below <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="0 Superscript number-sign"> <mml:semantics> <mml:msup> <mml:mn>0</mml:mn> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal"> # </mml:mi> </mml:mrow> </mml:msup> <mml:annotation encoding="application/x-tex">0^{\#}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . In a model obtained, there is a <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Sigma 2 Superscript 1"> <mml:semantics> <mml:msubsup> <mml:mi mathvariant="normal"> Σ </mml:mi> <mml:mn>2</mml:mn> <mml:mn>1</mml:mn> </mml:msubsup> <mml:annotation encoding="application/x-tex">\Sigma ^1_2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> set which is weakly <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="omega 2"> <mml:semantics> <mml:msub> <mml:mi> ω </mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:annotation encoding="application/x-tex">\omega _2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -universally Baire but not <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="omega 2"> <mml:semantics> <mml:msub> <mml:mi> ω </mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:annotation encoding="application/x-tex">\omega _2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -universally Baire.
Fetched live from OpenAlex and de-inverted. Abstracts are not stored in this database: the inverted indexes are 8.6 GB of the frame’s 9.3 GB of text, and the host has 13 GB free.
How this classification was reachedexpand
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 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.001 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.002 |
| 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 itClassification
machine, unvalidatedMachine predicted; a candidate call from one teacher head, 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".