Bibliographic record
Abstract
The Open Colouring Axiom implies that the measure algebra cannot be embedded into P(N)/fin.We also discuss errors in previous results on the embeddability of the measure algebra.Introduction.The aim of this paper is to prove the following result.Main Theorem.The Open Colouring Axiom implies that the measure algebra cannot be embedded into the Boolean algebra P(N)/fin.By "the measure algebra" we mean the quotient of the σ-algebra of Borel sets of the real line by the ideal of sets of measure zero.There are various reasons, besides sheer curiosity, why it is of interest to know whether the measure algebra can be embedded into P(N)/fin.One reason is that there is great interest in determining what the subalgebras of P(N)/fin are.One of the earliest and most influential results in this direction is Parovichenko's theorem from [14], which states that every Boolean algebra of size ℵ 1 can be embedded into P(N)/fin, with the obvious corollary that the Continuum Hypothesis (CH) implies that P(N)/fin is a universal Boolean algebra of size c: a Boolean algebra embeds into P(N)/fin iff it is of size c or less.It is therefore natural to ask how much of this universality remains without assumptions beyond ZFC.It has long been known that every σ-centered Boolean algebra embeds into P(N)/fin but the question for more general c.c.c.Boolean algebras has proven to be much more difficult-with the case of the measure algebra being seen as a touchstone.
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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 | 0.007 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.002 | 0.005 |
| Scholarly communication | 0.004 | 0.012 |
| Open science | 0.001 | 0.004 |
| Research integrity | 0.001 | 0.004 |
| 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".