Bibliographic record
Abstract
Abstract Model theory studies the class of models of a given theory. We have already encountered two theorems that tend in this direction: the completeness theorem and the powerful compactness theorem, both of which assert that, under certain conditions, this class is not empty. The central notion in this chapter and for the kind of model theory that we will develop here is the notion of elementary substructure. Intuitively, M is an elementary substructure of N if, obviously, M is a substructure of N and if, for every finite sequences of elements of M and for every property F[s] that is expressible by a first-order formula, it is equivalent to verify that s satisfies F in M or that s satisfies F in N. This notion will be our concern for the first two sections; the important results will be the Lowenheim-Skolem theorems and their corollaries which imply that a countable theory that has an infinite model must have infinite models of every infinite cardinality.
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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 | 0.000 |
| 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 | 0.000 | 0.000 |
| Research integrity | 0.001 | 0.000 |
| Insufficient payload (model declined to judge) | 0.005 | 0.001 |
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".