SOME RESULTS IN THE EXTENSION WITH A COHERENT SUSLIN TREE (Aspects of Descriptive Set Theory)
Bibliographic record
Abstract
We show that under PFA $(S)$ , the coherent Suslin tree $S$ (which is a witness of the axiom PFA $(S)$ ) forces that there are no $\omega_{2}$ -Aronszajn trees.We also determine the values of cardinal invariants of the continuum in this extension. INTRODUCTIONIn [20], Stevo Todor\v{c}evi\v{C} introduced the forcing axiom PFA $(S)$ , which says that there exists a coherent Suslin tree $S$ such that the forcing axiom holds for every proper forcing which preserves $S$ to be Suslin, that is, for every proper forcing $\mathbb{P}$ which preserves $S$ to be Suslin and $\aleph_{1^{-}}$ many dense subsets $D_{\alpha},$ $\alpha\in\omega_{1}$ , of $\mathbb{P}$ , there exists a filter on $\mathbb{P}$ which intersects all the $D_{\alpha}$ .PFA $(S)[S]$ denotes the forcing extension with the coherent Suslin tree $S$ which is a witness of PFA $(S)$ .Since the preser- vation of a Suslin tree by the proper forcing is closed under countable support iteration (due to Tadatoshi Miyamoto [15]), it is consistent relative to some large cardinal assumption that PFA $(S)$ holds.The first appearance of such a forcing axiom is in the paper [13] due to Paul B. Larson and Todor\v{c}evi\v{c}.In this paper, they introduced the weak version of PFA $(S)$ , called Souslin's Axiom (in which the properness is replaced by the cccness), and under this axiom, the coherent Suslin tree $S$ , which is a witness of the axiom, forces a weak fragment of Martin's Axiom.In [20], it is also proved that under PFA $(S),$ $S$ forces the open graph dichotomy () and the P-ideal dichotomy.Namely, many consequences of PFA are satisfied in the extension with $S$ under 2000 Mathematics Subject Classification.$03E50,03E05,03E35$ .
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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 | 0.003 | 0.003 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.002 |
| Bibliometrics | 0.003 | 0.003 |
| Science and technology studies | 0.003 | 0.006 |
| Scholarly communication | 0.003 | 0.007 |
| Open science | 0.001 | 0.003 |
| Research integrity | 0.001 | 0.003 |
| Insufficient payload (model declined to judge) | 0.006 | 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".