Bibliographic record
Abstract
The axiom of choice was first formulated by Zermelo in 1904 and used to prove his Well-Ordering theorem. The axiom was considered controversial because it introduced a highly nonconstructive aspect that differed from other axioms of set theory. For some time it was mainly used in the form of the Well-Ordering theorem (which is actually equivalent to the choice axiom). In this formulation, the axiom permits arguments by the so-called transfinite induction. For about the last 50 years it has usually been used in the form of the Maximal Principle of Zorn (published in the 1930s, though a version by Hausdorff was published earlier.) If nontrivial variation with respect to some category S of abstract sets and arbitrary mappings is allowed in a category of variable sets, the axiom of choice tends not to hold as we have seen in Exercises 4.54, 6.12, and Section 10.3. In fact, the axiom is valid in certain very special toposes of variable sets determined relative to a category of abstract sets and arbitrary mappings, as was mentioned in Section 4.6. Here we will demonstrate that Zorn's Maximal Principle is equivalent to the axiom of choice. We will show this for forms of the Maximal Principle that use both chains and directed (or filtered) posets. The latter form is more suitable for arguments made in mathematical practice, whereas it is the former that we will see as a direct consequence of the axiom of choice. In addition we will consider Hausdorff's Maximal Principle and some other consequences of the choice axiom. Our proofs of the maximal principles will follow directly from the famous Fixed- Point theorem of Bourbaki (B.15 below).
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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 | 0.002 | 0.003 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.001 |
| Bibliometrics | 0.001 | 0.002 |
| Science and technology studies | 0.002 | 0.010 |
| Scholarly communication | 0.004 | 0.006 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.002 | 0.003 |
| Insufficient payload (model declined to judge) | 0.016 | 0.003 |
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".