Compactness Theorem for Some Generalized Second-Order Language
Why this work is in the frame
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Bibliographic record
Abstract
For the first-order language the compactness theorem was proved by K. Gdel and A. I. Mal'cev in 1936 Unfortunately, for the usual second-order language the compactness theorem does not hold. Moreover, the method of ultraproducts is also inapplicable to second-order models. A possible way out of this situation is to refuse the most vulnerable place in the construction of ultraproducts connected with the factorization relatively an ultrafilter, i.e., to stay working with the ordinary non factorized product. It compels us instead of the single usual set-theoretical equality = to use several generalized equalities first and second for first and second orders, and instead of the single usual set-theoretical belonging to use several generalized belongings < second . Following that it is necessary to refuse the usual set-theoretical interpretation ((x 0 ), . . . , (x k )) (u) of the second basic (after equality) atomic formula (x 0 , . . . , x k )u and to replace it by the generalized interpretation ((x 0 ), . . . , (x k ))< (u), where x i i are variables of the first-order types i , u is a variable of the second-order type = [ 0 , . . . , k ] (i.e. predicate), and is some evaluation of variables on some mathematical system U.
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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 | 0.010 | 0.002 |
| 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 | 0.002 | 0.000 |
| 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