{ "model": { "rank": 9592, "code": "thing-q191780", "model_id": "vr.tr.ordinal-number", "name": "ordinal number", "purpose": "Let an agent explain ordinal numbers in set theory and in everyday and statistical use, distinguish ordinal from cardinal numbers, support reasoning with transfinite ordinals and ordinal data, and support learning.", "family": "Thing Registry", "category": "Cross-cutting context", "status": "research-draft", "kind": "thing", "plane": "XCT", "domain": "XCT.QLT", "industry": "", "version": "", "url": "/models/thing/q191780/", "tier": 2, "score": 47, "payload": { "layer": "wikidata", "aliases": [ "ordinal variable", "Hartogs number", "successor ordinal", "singular ordinal", "epsilon numbers", "tax number", "nimber", "limit ordinal", "large countable ordinal", "von Neumann ordinal", "opus", "critical point", "recursive ordinal", "regular ordinal", "admissible ordinal", "WoO" ], "aliasCount": 16, "merged": 16, "knownIn": 47, "facets": null, "markers": [], "lexicalClass": "", "senseRank": null, "alsoRegisteredAs": null, "source": { "dataset": "wikidata", "item": "Q191780", "url": "https://www.wikidata.org/wiki/Q191780", "license": "CC0 1.0" } }, "research": { "vercy": "1.0-draft", "publication": { "status": "research-draft", "adjudicationStatus": "unreviewed", "publishableCanonical": false, "generatedAt": "2026-09-12T03:36:19Z", "providers": [ "Claude" ], "breadth": "written by Claude from model knowledge without web access - no source was read, every claim is a lead to verify", "pass": 2, "wave": 2, "engine": "claude" }, "metaModel": { "id": "THING-Q191780", "registryId": "vr.tr.ordinal-number", "name": "ordinal number", "version": "0.2.0-wave.2", "entryKind": "thing", "family": "Thing Registry", "domain": [ "XCT.QLT" ], "status": "research-draft" }, "canonicalUrl": "https://ver.cy/models/thing/q191780/", "model": { "registry_id": "vr.tr.ordinal-number", "name": "ordinal number", "purpose": "Let an agent explain ordinal numbers in set theory and in everyday and statistical use, distinguish ordinal from cardinal numbers, support reasoning with transfinite ordinals and ordinal data, and support learning.", "definition": "In set theory, a number describing the order type of a well-ordered set, extending the natural numbers through the first infinite ordinal omega and beyond by successor and limit ordinals, with special ordinals such as epsilon numbers, Hartogs numbers and singular ordinals; in everyday and statistical use, an ordinal number or ordinal variable expresses position or rank such as first, second and third, and in administration ordinal identifiers such as tax numbers are sometimes so called.", "what_it_is_for": "Numbers expressing order position.", "affordances": [ "explain set-theoretic ordinals", "explain everyday and statistical ordinals", "distinguish from cardinals", "support learning" ], "distinguishing_features": [ "Order types", "Transfinite extension", "Successor and limit", "Everyday rank sense" ], "appearance": "Not physical; numbers and order types.", "visual_identification": [ "Order type of a well-ordered set or a rank position", "First, second, omega and beyond", "Cardinal numbers count size; nominal labels have no order" ], "physical_properties": [], "families_and_kinds": [ "finite ordinals", "transfinite ordinals from omega upward", "successor and limit ordinals", "special ordinals such as epsilon and Hartogs numbers", "ordinal variables in statistics" ], "related_models": [ { "relation": "is a kind of", "target": "number", "why": "category" }, { "relation": "is a kind of", "target": "quantity", "why": "category" }, { "relation": "is related to", "target": "cardinal", "why": "the counting counterpart" }, { "relation": "is related to", "target": "real", "why": "another number class" } ], "identifiers": [], "standards_and_regulation": [ "Set-theoretic conventions", "Statistical scale definitions", "No regulation" ], "failure_modes_and_hazards": [ "Confusing ordinal and cardinal arithmetic", "Treating ordinal data as interval data", "Mixing set-theoretic and everyday senses" ], "in_scope": [], "out_of_scope": [], "characteristics": [] }, "sources": [], "structure": { "bundles": [ { "id": "understand", "name": "Understand", "description": "What ordinal numbers are.", "rationale": "Mathematics.", "layers": [ { "id": "settheory", "name": "Set theory", "description": "Ordinals in set theory.", "findings": [ { "id": "settheory-finding", "name": "Set theory", "description": "Set theory.", "questions": [ { "text": "How are ordinals defined as order types or von Neumann ordinals, and how do successor and limit ordinals extend beyond omega?", "kind": "definition" }, { "text": "Is the set-theoretic or the everyday sense meant?", "kind": "boundary" } ] } ] }, { "id": "everyday", "name": "Everyday", "description": "Everyday and statistical ordinals.", "findings": [ { "id": "everyday-finding", "name": "Everyday", "description": "Everyday.", "questions": [ { "text": "What are ordinal numbers as ranks and ordinal variables in statistics?", "kind": "definition" }, { "text": "Which entry fits levels of measurement?", "kind": "action" } ] } ] } ] }, { "id": "reason", "name": "Reason", "description": "Reasoning with ordinals.", "rationale": "Practice.", "layers": [ { "id": "arithmetic", "name": "Arithmetic", "description": "Ordinal arithmetic.", "findings": [ { "id": "arithmetic-finding", "name": "Arithmetic", "description": "Arithmetic.", "questions": [ { "text": "How do ordinal addition, multiplication and exponentiation work, and why are they not commutative?", "kind": "definition" }, { "text": "Which entry fits transfinite induction?", "kind": "action" } ] } ] }, { "id": "data", "name": "Data", "description": "Ordinal data.", "findings": [ { "id": "data-finding", "name": "Data", "description": "Data.", "questions": [ { "text": "How should ordinal data be analysed and which statistics are appropriate?", "kind": "action" }, { "text": "Which entry fits statistics?", "kind": "action" } ] } ] } ] }, { "id": "special", "name": "Special", "description": "Special ordinals.", "rationale": "Theory.", "layers": [ { "id": "named", "name": "Named", "description": "Named ordinals.", "findings": [ { "id": "named-finding", "name": "Named", "description": "Named.", "questions": [ { "text": "What are epsilon numbers, Hartogs numbers and singular ordinals?", "kind": "definition" }, { "text": "Which entry fits large ordinals?", "kind": "action" } ] } ] }, { "id": "cardinals", "name": "Cardinals", "description": "Ordinals and cardinals.", "findings": [ { "id": "cardinals-finding", "name": "Cardinals", "description": "Cardinals.", "questions": [ { "text": "How do ordinals relate to cardinals and the axiom of choice?", "kind": "definition" }, { "text": "Which entry fits cardinal numbers?", "kind": "action" } ] } ] } ] }, { "id": "learn", "name": "Learn", "description": "History and teaching.", "rationale": "Education.", "layers": [ { "id": "history", "name": "History", "description": "History.", "findings": [ { "id": "history-finding", "name": "History", "description": "History.", "questions": [ { "text": "How did Cantor and von Neumann develop ordinal numbers?", "kind": "provenance" }, { "text": "Which references are standard?", "kind": "provenance" } ] } ] }, { "id": "teach", "name": "Teach", "description": "Teaching.", "findings": [ { "id": "teach-finding", "name": "Teach", "description": "Teaching.", "questions": [ { "text": "How can ordinals be taught in language and in mathematics?", "kind": "action" }, { "text": "Which misconceptions arise?", "kind": "provenance" } ] } ] } ] } ] }, "openQuestions": [ "Should transfinite ordinals be a separate entry?", "How should set theory resources be linked?", "How should statistical guidance be linked?" ], "statistics": { "bundles": 4, "layers": 8, "findings": 8, "questions": 16 } }, "draft": { "generator": "vr.draft.v3", "status": "draft-generated", "researched": false, "archetype": "abstract concept", "method": "Written from the archetype playbook - what this kind of thing needs beyond identity and provenance - and from the structure that recurred across 6,333 models already researched by two engines. Applied to this entry by rule. No source was read for this thing and no claim here is researched. This entry carries no facets of its own, so they were inferred from its domain - a guess about a whole domain applied to one thing.", "facetsInferred": true, "nextPass": "A researcher replaces this draft with a sourced specification. Treat every sentence below as a proposal to argue with.", "purpose": "Give an agent a durable, checkable way to recognise a ordinal number, record what state it is in, and decide what may be done with it.", "whatItIs": "Let an agent explain ordinal numbers in set theory and in everyday and statistical use, distinguish ordinal from cardinal numbers, support reasoning with transfinite ordinals and ordinal data, and support learning.", "characteristics": { "substance": "abstract", "origin": "conceptual", "agency": "inert" }, "whatYouCanDoWithIt": [ "observed and measured" ], "distinguishingFeatures": [ "Names folded into this entry, which a task may need to split apart again: ordinal variable, Hartogs number, successor ordinal, singular ordinal, epsilon numbers, tax number, nimber, limit ordinal, large countable ordinal, von Neumann ordinal, opus, critical point.", "16 finer distinctions are held as aliases rather than separate entries, because telling them apart needs a task that asks for it.", "Described in 47 Wikipedia languages, which is a measure of how widely the thing is known, not of how important it is." ], "openQuestionsForResearch": [ "Which of the bundles below does a real task actually need, and which are ceremony?", "What does this thing have that the facets do not capture at all?", "Which neighbouring kind is most often confused with a ordinal number, and on what evidence are they told apart?" ], "whatItIsMadeOf": "an abstraction with no physical instance", "physicalCharacter": [ "Does nothing on its own; everything it does, something else did to it.", "These come from the domain this entry sits in rather than from the entry itself, so treat them as a first guess about the whole domain applied to one thing." ], "whatCanBeDoneWithIt": [ "observe it, measure it, record its state" ], "howItIsRecognised": [ "Nothing to see. What is recognised is an instance of it, and which instances count is exactly what is argued about." ], "relatedModels": [ { "relation": "covers", "note": "Finer kinds folded into this entry because telling them apart needs a task that asks for it. Each is a model waiting to be split out when one does.", "targets": [ "ordinal variable", "Hartogs number", "successor ordinal", "singular ordinal", "epsilon numbers", "tax number", "nimber", "limit ordinal", "large countable ordinal", "von Neumann ordinal", "opus", "critical point" ] } ], "standing": "Described in 47 Wikipedia languages, which measures how widely it is written about rather than how important or how common it is. 16 finer distinctions are held inside this entry as names rather than as separate models.", "structure": { "bundles": [ { "id": "identity-and-classification", "name": "Identity, naming and classification", "description": "How an agent tells one ordinal number from another, and a ordinal number from things that resemble it.", "rationale": "Recognition comes before every other claim. Without stable identity nothing else in the model can be trusted to be about the same thing twice.", "layers": [ { "id": "naming-and-identifiers", "name": "Names and identifiers", "description": "The names this thing goes by and the identifiers that survive translation and time.", "findings": [ { "id": "preferred-name-and-aliases", "name": "Preferred name, aliases and local names", "description": "Which name to use, which names mean the same thing, and which merely sound similar.", "questions": [ { "id": "preferred-name-and-aliases-q01", "text": "What identifies and describes the name of a ordinal number, and in what units or vocabulary?", "kind": "definition", "answer_data": [ "identifiers", "types and classes", "values with units", "explicit unknowns" ] }, { "id": "preferred-name-and-aliases-q02", "text": "Who or what asserted this about the name of a ordinal number, by which method, and when was it true?", "kind": "provenance", "answer_data": [ "authority", "method", "evidence", "event time", "knowledge time" ] }, { "id": "preferred-name-and-aliases-q03", "text": "What may an agent decide or do once the name of a ordinal number is known, and what must it refuse?", "kind": "action", "answer_data": [ "permitted actions", "preconditions", "refusals", "escalation" ] } ] }, { "id": "stable-identifiers", "name": "Stable identifiers and external keys", "description": "Identifiers that keep pointing at this kind of thing across systems and languages.", "questions": [ { "id": "stable-identifiers-q01", "text": "What identifies and describes an identifier for a ordinal number, and in what units or vocabulary?", "kind": "definition", "answer_data": [ "identifiers", "types and classes", "values with units", "explicit unknowns" ] }, { "id": "stable-identifiers-q02", "text": "Who or what asserted this about an identifier for a ordinal number, by which method, and when was it true?", "kind": "provenance", "answer_data": [ "authority", "method", "evidence", "event time", "knowledge time" ] }, { "id": "stable-identifiers-q03", "text": "What may an agent decide or do once an identifier for a ordinal number is known, and what must it refuse?", "kind": "action", "answer_data": [ "permitted actions", "preconditions", "refusals", "escalation" ] } ] } ] }, { "id": "classification-and-granularity", "name": "Classification and granularity", "description": "Where a ordinal number sits among kinds, and how finely a task needs to cut it.", "findings": [ { "id": "kind-and-parents", "name": "Kind, parents and neighbouring kinds", "description": "The classes this thing belongs to and the ones it is next to.", "questions": [ { "id": "kind-and-parents-q01", "text": "What identifies and describes the kind of a ordinal number, and in what units or vocabulary?", "kind": "definition", "answer_data": [ "identifiers", "types and classes", "values with units", "explicit unknowns" ] }, { "id": "kind-and-parents-q02", "text": "Who or what asserted this about the kind of a ordinal number, by which method, and when was it true?", "kind": "provenance", "answer_data": [ "authority", "method", "evidence", "event time", "knowledge time" ] }, { "id": "kind-and-parents-q03", "text": "What may an agent decide or do once the kind of a ordinal number is known, and what must it refuse?", "kind": "action", "answer_data": [ "permitted actions", "preconditions", "refusals", "escalation" ] } ] }, { "id": "distinguishing-features", "name": "Distinguishing features", "description": "What separates a ordinal number from the things most often confused with it.", "questions": [ { "id": "distinguishing-features-q01", "text": "What identifies and describes what distinguishes a ordinal number, and in what units or vocabulary?", "kind": "definition", "answer_data": [ "identifiers", "types and classes", "values with units", "explicit unknowns" ] }, { "id": "distinguishing-features-q02", "text": "Who or what asserted this about what distinguishes a ordinal number, by which method, and when was it true?", "kind": "provenance", "answer_data": [ "authority", "method", "evidence", "event time", "knowledge time" ] }, { "id": "distinguishing-features-q03", "text": "What may an agent decide or do once what distinguishes a ordinal number is known, and what must it refuse?", "kind": "action", "answer_data": [ "permitted actions", "preconditions", "refusals", "escalation" ] } ] } ] } ] }, { "id": "state-and-lifecycle", "name": "State and lifecycle", "description": "The states a ordinal number passes through and the events that move it between them.", "rationale": "Most decisions about a thing depend on what state it is in now, which is a claim with a time on it, not a property.", "layers": [ { "id": "lifecycle-stages", "name": "Lifecycle stages", "description": "From coming into existence to ceasing to be one of these.", "findings": [ { "id": "stages-and-transitions", "name": "Stages and transitions", "description": "The stages worth naming and what moves a ordinal number between them.", "questions": [ { "id": "stages-and-transitions-q01", "text": "What identifies and describes the lifecycle of a ordinal number, and in what units or vocabulary?", "kind": "definition", "answer_data": [ "identifiers", "types and classes", "values with units", "explicit unknowns" ] }, { "id": "stages-and-transitions-q02", "text": "Who or what asserted this about the lifecycle of a ordinal number, by which method, and when was it true?", "kind": "provenance", "answer_data": [ "authority", "method", "evidence", "event time", "knowledge time" ] }, { "id": "stages-and-transitions-q03", "text": "What may an agent decide or do once the lifecycle of a ordinal number is known, and what must it refuse?", "kind": "action", "answer_data": [ "permitted actions", "preconditions", "refusals", "escalation" ] } ] } ] }, { "id": "observations-and-status", "name": "Observations and current status", "description": "What is observed about a ordinal number, how often and by whom.", "findings": [ { "id": "observation-record", "name": "Observation record", "description": "How an observation of a ordinal number is recorded so that it can be superseded rather than overwritten.", "questions": [ { "id": "observation-record-q01", "text": "What identifies and describes an observation of a ordinal number, and in what units or vocabulary?", "kind": "definition", "answer_data": [ "identifiers", "types and classes", "values with units", "explicit unknowns" ] }, { "id": "observation-record-q02", "text": "Who or what asserted this about an observation of a ordinal number, by which method, and when was it true?", "kind": "provenance", "answer_data": [ "authority", "method", "evidence", "event time", "knowledge time" ] }, { "id": "observation-record-q03", "text": "What may an agent decide or do once an observation of a ordinal number is known, and what must it refuse?", "kind": "action", "answer_data": [ "permitted actions", "preconditions", "refusals", "escalation" ] } ] } ] } ] }, { "id": "definitions-in-contest", "name": "Definitions and who holds them", "description": "What ordinal number is taken to mean, and by whom.", "rationale": "When a field disagrees about a concept, the disagreement is the content. A model that picks one definition silently destroys the information.", "layers": [ { "id": "competing-definitions", "name": "Competing definitions", "description": "The main readings and the traditions behind them.", "findings": [ { "id": "definition-map", "name": "Definitions and their holders", "description": "Each definition with the school or body that holds it.", "questions": [ { "id": "definition-map-q01", "text": "Which definitions of ordinal number are in use, and which tradition or body holds each?", "kind": "definition" }, { "id": "definition-map-q02", "text": "What turns on the difference between them in practice?", "kind": "boundary" } ] } ] }, { "id": "operationalisation", "name": "Operationalisation", "description": "How it is measured or applied when it has to be.", "findings": [ { "id": "operational-record", "name": "Measures and proxies", "description": "Instruments and indicators used to stand in for it.", "questions": [ { "id": "operational-record-q01", "text": "How is ordinal number operationalised or measured in practice, and by what instrument?", "kind": "measurement" }, { "id": "operational-record-q02", "text": "What does that operationalisation leave out, and when does that matter?", "kind": "boundary" } ] } ] } ] }, { "id": "instances-and-use", "name": "Instances, use and consequence", "description": "What counts as an instance of ordinal number and what follows from calling something that.", "rationale": "Applying a concept is an act with consequences, so a model must say what the label licenses and what it does not.", "layers": [ { "id": "instances", "name": "What counts as an instance", "description": "Clear cases, borderline cases and non-cases.", "findings": [ { "id": "instance-tests", "name": "Tests for an instance", "description": "What would settle whether something falls under it.", "questions": [ { "id": "instance-tests-q01", "text": "What would settle whether something is an instance of ordinal number?", "kind": "boundary" }, { "id": "instance-tests-q02", "text": "Which borderline cases are argued about, and on what grounds?", "kind": "definition" } ] } ] }, { "id": "consequence", "name": "Consequence of application", "description": "Rights, duties or decisions that follow from the label.", "findings": [ { "id": "consequence-record", "name": "What the label licenses", "description": "What an agent may do once something is classified this way.", "questions": [ { "id": "consequence-record-q01", "text": "What follows practically once something is treated as ordinal number?", "kind": "action" }, { "id": "consequence-record-q02", "text": "What must an agent not infer from the label alone?", "kind": "action" } ] } ] } ] }, { "id": "provenance-and-evidence", "name": "Provenance, evidence and time", "description": "Where every claim about a ordinal number came from and when it held.", "rationale": "A claim without a source and a time cannot be superseded, only overwritten, and an agent that overwrites loses the ability to explain itself.", "layers": [ { "id": "source-and-authority", "name": "Source and authority", "description": "Who said it, on what evidence, and how strongly.", "findings": [ { "id": "claim-provenance", "name": "Claim provenance and confidence", "description": "The authority behind each claim about a ordinal number and how confident it is.", "questions": [ { "id": "claim-provenance-q01", "text": "What identifies and describes a claim about a ordinal number, and in what units or vocabulary?", "kind": "definition", "answer_data": [ "identifiers", "types and classes", "values with units", "explicit unknowns" ] }, { "id": "claim-provenance-q02", "text": "Who or what asserted this about a claim about a ordinal number, by which method, and when was it true?", "kind": "provenance", "answer_data": [ "authority", "method", "evidence", "event time", "knowledge time" ] }, { "id": "claim-provenance-q03", "text": "What may an agent decide or do once a claim about a ordinal number is known, and what must it refuse?", "kind": "action", "answer_data": [ "permitted actions", "preconditions", "refusals", "escalation" ] } ] } ] }, { "id": "time-and-versions", "name": "Time, versions and supersession", "description": "When a claim was true, when it was learnt, and what replaced it.", "findings": [ { "id": "validity-and-supersession", "name": "Validity period and supersession", "description": "How an old claim about a ordinal number is retired without being erased.", "questions": [ { "id": "validity-and-supersession-q01", "text": "What identifies and describes the validity of a claim about a ordinal number, and in what units or vocabulary?", "kind": "definition", "answer_data": [ "identifiers", "types and classes", "values with units", "explicit unknowns" ] }, { "id": "validity-and-supersession-q02", "text": "Who or what asserted this about the validity of a claim about a ordinal number, by which method, and when was it true?", "kind": "provenance", "answer_data": [ "authority", "method", "evidence", "event time", "knowledge time" ] }, { "id": "validity-and-supersession-q03", "text": "What may an agent decide or do once the validity of a claim about a ordinal number is known, and what must it refuse?", "kind": "action", "answer_data": [ "permitted actions", "preconditions", "refusals", "escalation" ] } ] } ] } ] } ] }, "statistics": { "bundles": 5, "layers": 10, "findings": 12, "questions": 32 } } } }