Bibliographic record
Abstract
Summary. Theorem (5) states that if an iterate of a function has a unique fixpoint then it is also the fixpoint of the function. It has been included here in response to P. Andrews claim that such a proof in set theory takes thousands of lines when one starts with the axioms. While probably true, such a claim is misleading about the usefulness of proof-checking systems based on set theory. Next, we prove the existence of the least and the greatest fixpoints for ⊆-monotone functions from a powerset to a powerset of a set. Scheme Knaster is the Knaster theorem about the existence of fixpoints, cf. [14]. Theorem (11) is the Banach decomposition theorem which is then used to prove the Schröder-Bernstein theorem (12) (we followed Paulson’s development of these theorems in Isabelle [16]). It is interesting to note that the last theorem when stated in Mizar in terms of cardinals becomes trivial to prove as in the Mizar development of cardinals the ≤ relation is synonymous with ⊆. Section 3 introduces the notion of the lattice of a lattice subset provided the subset has lubs and glbs. The main theorem of Section 4 is the Tarski theorem (43) that every monotone function f over a complete lattice L has a complete lattice of fixpoints. As the consequence of this theorem we get the existence of the least fixpoint equal to f β (⊥L) for some ordinal β with cardinality not bigger than the cardinality of the carrier of L, cf. [14], and analogously the existence of the greatest fixpoint equal to f β (⊤L). Section 5 connects the fixpoint properties of monotone functions over complete lattices with the fixpoints of ⊆-monotone functions over the lattice of subsets of a set (Boolean lattice).
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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) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.002 | 0.002 |
| Science and technology studies | 0.002 | 0.001 |
| Scholarly communication | 0.007 | 0.003 |
| Open science | 0.001 | 0.003 |
| Research integrity | 0.002 | 0.002 |
| Insufficient payload (model declined to judge) | 0.741 | 0.438 |
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".