The real line in elementary submodels of set theory
Why this work is in the frame
A frame that forgets how it found something cannot be audited. These are the routes that admitted this work.
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.
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.
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 | 0.003 | 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 | 0.002 | 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