Bibliographic record
Abstract
The infinite in mathematics has two manifestations. Its occurrence in analysis has been satisfactorily formalized and demystified by the -δ method of Bolzano, Cauchy and Weierstrass. It is of course the ‘set-theoretic infinite ’ that concerns me here. Once the existence of an infinite set is ac-cepted, the axioms of set theory imply the existence of a transfinite hierar-chy of larger and larger orders of infinity. I shall review some well-known facts about the influence of these axioms of infinity ([28]) to the everyday mathematical practice and point out to some, as of yet not understood, phe-nomena at the level of the third-order arithmetic. Technical details from both set theory and operator algebras are kept at the bare minimum. In the Appendix I include definitions of arithmetical and analytical hierarchies in order to make this paper more accessible to non-logicians. In this paper I am taking a position intermediate between pluralism and non-pluralism (as defined in [34]) with an eye for applications outside of set theory. Acknowledgments This paper is partly based on my talks at the ‘Truth and Infinity ’ work-shop (IMS, 2011) and the ‘Connes Embedding Problem ’ workshop (Ottawa, 2008). I would like to thank the organizers of both meetings. Another driv-ing force for this paper—and much of my work—originated in conversations with functional analysts, too numerous to list here, over the past several
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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 | 0.008 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.002 | 0.001 |
| Science and technology studies | 0.002 | 0.021 |
| Scholarly communication | 0.005 | 0.017 |
| Open science | 0.001 | 0.004 |
| Research integrity | 0.001 | 0.003 |
| Insufficient payload (model declined to judge) | 0.005 | 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".