Bibliographic record
Abstract
The use of elementary submodels has become a standard tool in set-theoretic topology and infinitary combinatorics. Thus, in studying some combinatorial objects, one embeds them in a set, M , which is an elementary submodel of the universe, V (that is, ( M ; Є) ≺ ( V ; Є)). Applying the downward Löwenheim-Skolem Theorem, one can bound the cardinality of M . This tool enables one to capture various complicated closure arguments within the simple “≺”. However, in this paper, as in the paper [JT], we study the tool for its own sake. [JT] discussed various general properties of topological spaces in elementary submodels. In this paper, we specialize this consideration to the space of real numbers, ℝ. Our models M are not in general transitive. We will always have ℝ Є M , but not usually ℝ ⊆ M . We plan to study properties of the ℝ ⋂ M 's. In particular, as M varies, we wish to study whether any two of these ℝ ⋂ M 's are isomorphic as topological spaces, linear orders, or fields. As usual, it takes some sleight-of-hand to formalize these notions within the standard axioms of set theory (ZFC), since within ZFC, one cannot actually define the notion ( M ;Є) ≺ ( V ;Є). Instead, one proves theorems about M such that ( M ;Є) ≺ ( H (θ);Є), where θ is a “large enough” cardinal; here, H (θ) is the collection of all sets whose transitive closure has size less than θ.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.004 | 0.004 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.002 | 0.002 |
| Science and technology studies | 0.002 | 0.009 |
| Scholarly communication | 0.006 | 0.013 |
| Open science | 0.001 | 0.003 |
| Research integrity | 0.001 | 0.004 |
| Insufficient payload (model declined to judge) | 0.004 | 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 source (direct Gemma or distilled Codex), 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".