Club-Isomorphisms of Aronszajn Trees in the Extension with a Suslin Tree
Bibliographic record
Abstract
We show that, under PFA.S/, a coherent Suslin tree forces that every two Aronszajn trees are club-isomorphic. .Todorčević [27] introduced the proper forcing axiom PFA.S/.This axiom asserts that there exists a coherent Suslin tree S such that the forcing axiom holds for every proper forcing notion P which preserves that S is Suslin, that is, for any set ¹D ˛I ˛2 ! 1 º of @ 1 -many dense open subsets of P, there exists a filter of P which intersects all of the D ˛'s.We note that, under the existence of a supercompact cardinal, PFA.S/ can be forced by the use of the theorem due to Miyamoto [16, Theorem 1.3] (see Theorem 2.1 below). Introduction SLarson and Todorčević [11] introduced the forcing axiom MA @ 1 .S/, which is analogous to PFA.S/ replacing "proper" with "countable chain condition" (ccc), to give the consistency of the affirmative answer to Katětov's problem.In particular, they introduced the axiom K 2 .rec/,which is a fragment of MA @ 1 , and proved that K 2 .rec/holds in the extension with a coherent Suslin tree S (which witnesses the axiom MA @ 1 .S/).Later, Todorčević [27] proved that, under PFA.S/, a coherent Suslin tree S (which witnesses PFA.S/) forces that every compact hereditarily normal space satisfying the countable chain condition is hereditarily separable and hereditarily Lindelöf.
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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 | 0.001 | 0.002 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.002 | 0.001 |
| Science and technology studies | 0.003 | 0.002 |
| Scholarly communication | 0.002 | 0.004 |
| Open science | 0.000 | 0.004 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.004 | 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 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".