{ "model": { "rank": 7582, "code": "thing-q163875", "model_id": "vr.tr.cardinal-number", "name": "cardinal number", "purpose": "Let an agent explain cardinal numbers in both the everyday and the set-theoretic sense, distinguish cardinals from ordinals, relay the basic results on infinite cardinals with attribution, and route advanced set theory to specialist references.", "family": "Thing Registry", "category": "Cross-cutting context", "status": "research-draft", "kind": "thing", "plane": "XCT", "domain": "XCT.QLT", "industry": "", "version": "", "url": "/models/thing/q163875/", "tier": 2, "score": 54, "payload": { "layer": "wikidata", "aliases": [ "regular cardinal", "large cardinal", "limit cardinal", "Berkeley cardinal", "Löwenheim number", "count variable", "shrewd cardinal", "superstrong cardinal", "unfoldable cardinal", "beth number", "gimel number", "daleth number", "aleph number", "weakly inaccessible cardinal", "ineffable cardinal", "indescribable cardinal", "iterable cardinal", "worldly cardinal", "Woodin cardinal", "Erdős cardinal", "Jónsson cardinal", "reflecting cardinal", "Reinhardt cardinal", "remarkable cardinal", "Shelah cardinal", "subcompact cardinal", "subtle cardinal", "Suslin cardinal", "tall cardinal" ], "aliasCount": 29, "merged": 29, "knownIn": 54, "facets": null, "markers": [], "lexicalClass": "", "senseRank": null, "alsoRegisteredAs": null, "source": { "dataset": "wikidata", "item": "Q163875", "url": "https://www.wikidata.org/wiki/Q163875", "license": "CC0 1.0" } }, "research": { "vercy": "1.0-draft", "publication": { "status": "research-draft", "adjudicationStatus": "unreviewed", "publishableCanonical": false, "generatedAt": "2026-09-12T20:42:23Z", "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": 3, "engine": "claude" }, "metaModel": { "id": "THING-Q163875", "registryId": "vr.tr.cardinal-number", "name": "cardinal number", "version": "0.2.0-wave.3", "entryKind": "thing", "family": "Thing Registry", "domain": [ "XCT.QLT" ], "status": "research-draft" }, "canonicalUrl": "https://ver.cy/models/thing/q163875/", "model": { "registry_id": "vr.tr.cardinal-number", "name": "cardinal number", "purpose": "Let an agent explain cardinal numbers in both the everyday and the set-theoretic sense, distinguish cardinals from ordinals, relay the basic results on infinite cardinals with attribution, and route advanced set theory to specialist references.", "definition": "A number that answers how many: the size of a set, from the finite cardinals zero, one, two and so on to the infinite cardinals of set theory such as aleph-null, the size of the natural numbers, and the cardinality of the continuum; in grammar, a cardinal number is a counting word as opposed to an ordinal like first or second.", "what_it_is_for": "Measuring the size of collections.", "affordances": [ "explain cardinals versus ordinals", "explain finite and infinite cardinals", "relay basic theorems such as Cantor theorem", "route large cardinal questions to specialist references" ], "distinguishing_features": [ "Size of a set", "Comparison by bijection", "Infinite cardinals beyond the finite", "Grammatical sense as counting words" ], "appearance": "Not visible; numerals and set-theoretic symbols such as aleph.", "visual_identification": [ "Answers how many", "Infinite cardinals written with aleph and beth symbols", "Ordinals answer which position and have a different arithmetic" ], "physical_properties": [], "families_and_kinds": [ "finite cardinals", "countable and uncountable infinite cardinals", "aleph numbers and beth numbers", "regular, singular and limit cardinals", "large cardinals as axioms", "cardinal numerals in grammar" ], "related_models": [ { "relation": "is a kind of", "target": "number", "why": "a kind of number" }, { "relation": "is contrasted with", "target": "ordinal number", "why": "position rather than size" }, { "relation": "is defined by", "target": "set theory", "why": "via bijections" }, { "relation": "is related to", "target": "infinity", "why": "infinite cardinals" } ], "identifiers": [], "standards_and_regulation": [ "No regulation; definitions follow ZFC set theory and standard texts" ], "failure_modes_and_hazards": [ "Confusing cardinals with ordinals", "Treating infinite cardinals as ordinary numbers", "Presenting independent statements such as the continuum hypothesis as settled" ], "in_scope": [], "out_of_scope": [], "characteristics": [] }, "sources": [], "structure": { "bundles": [ { "id": "concept", "name": "Concept", "description": "What a cardinal is.", "rationale": "Definition.", "layers": [ { "id": "definition", "name": "Definition", "description": "Definition.", "findings": [ { "id": "definition-finding", "name": "Definition", "description": "Definition.", "questions": [ { "text": "What is a cardinal number, and how is the size of a set compared by bijection?", "kind": "definition" }, { "text": "Is the question about cardinals, ordinals or counting words in grammar?", "kind": "boundary" } ] } ] }, { "id": "finite", "name": "Finite", "description": "Finite cardinals.", "findings": [ { "id": "finite-finding", "name": "Finite", "description": "Finite.", "questions": [ { "text": "How are finite cardinals defined and used in counting?", "kind": "definition" }, { "text": "Which entry fits natural numbers?", "kind": "action" } ] } ] } ] }, { "id": "infinite", "name": "Infinite", "description": "Infinite cardinals.", "rationale": "Theory.", "layers": [ { "id": "countable", "name": "Countable", "description": "Countable and uncountable.", "findings": [ { "id": "countable-finding", "name": "Countable", "description": "Countable.", "questions": [ { "text": "What are aleph-null and the continuum, and what does Cantor theorem show?", "kind": "definition" }, { "text": "Which entry fits Cantor diagonal argument?", "kind": "action" } ] } ] }, { "id": "arithmetic", "name": "Arithmetic", "description": "Cardinal arithmetic.", "findings": [ { "id": "arithmetic-finding", "name": "Arithmetic", "description": "Arithmetic.", "questions": [ { "text": "How do addition, multiplication and exponentiation work for infinite cardinals?", "kind": "provenance" }, { "text": "Which references are standard?", "kind": "provenance" } ] } ] } ] }, { "id": "advanced", "name": "Advanced", "description": "Set theory frontiers.", "rationale": "Rigour.", "layers": [ { "id": "independence", "name": "Independence", "description": "Continuum hypothesis.", "findings": [ { "id": "independence-finding", "name": "Independence", "description": "Independence.", "questions": [ { "text": "What is the continuum hypothesis, and what does its independence from ZFC mean?", "kind": "provenance" }, { "text": "What is proven versus undecidable?", "kind": "boundary" } ] } ] }, { "id": "large", "name": "Large", "description": "Large cardinals.", "findings": [ { "id": "large-finding", "name": "Large", "description": "Large.", "questions": [ { "text": "What are large cardinal axioms, in outline, and why are they studied?", "kind": "provenance" }, { "text": "Which entry fits large cardinals?", "kind": "action" } ] } ] } ] }, { "id": "learn", "name": "Learn", "description": "Grammar, history and teaching.", "rationale": "Context.", "layers": [ { "id": "grammar", "name": "Grammar", "description": "Cardinal numerals.", "findings": [ { "id": "grammar-finding", "name": "Grammar", "description": "Grammar.", "questions": [ { "text": "How do languages express cardinal numerals, and how do they differ from ordinals?", "kind": "provenance" }, { "text": "Which entry fits numerals in grammar?", "kind": "action" } ] } ] }, { "id": "history", "name": "History", "description": "History and teaching.", "findings": [ { "id": "history-finding", "name": "History", "description": "History.", "questions": [ { "text": "How did Cantor develop the theory of cardinals, and how is it taught?", "kind": "provenance" }, { "text": "Which misconceptions arise?", "kind": "provenance" } ] } ] } ] } ] }, "openQuestions": [ "Should large cardinals be a separate entry?", "How should set theory references be linked?", "The registry entry has merged aliases for specific large cardinals and a grammatical sense; should they be split off?" ], "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 cardinal number, record what state it is in, and decide what may be done with it.", "whatItIs": "Let an agent explain cardinal numbers in both the everyday and the set-theoretic sense, distinguish cardinals from ordinals, relay the basic results on infinite cardinals with attribution, and route advanced set theory to specialist references.", "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: regular cardinal, large cardinal, limit cardinal, Berkeley cardinal, Löwenheim number, count variable, shrewd cardinal, superstrong cardinal, unfoldable cardinal, beth number, gimel number, daleth number.", "29 finer distinctions are held as aliases rather than separate entries, because telling them apart needs a task that asks for it.", "Described in 54 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 cardinal 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": [ "regular cardinal", "large cardinal", "limit cardinal", "Berkeley cardinal", "Löwenheim number", "count variable", "shrewd cardinal", "superstrong cardinal", "unfoldable cardinal", "beth number", "gimel number", "daleth number" ] } ], "standing": "Described in 54 Wikipedia languages, which measures how widely it is written about rather than how important or how common it is. 29 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 cardinal number from another, and a cardinal 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 cardinal 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 cardinal 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 cardinal 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 cardinal 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 cardinal 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 cardinal 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 cardinal 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 cardinal 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 cardinal 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 cardinal 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 cardinal number from the things most often confused with it.", "questions": [ { "id": "distinguishing-features-q01", "text": "What identifies and describes what distinguishes a cardinal 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 cardinal 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 cardinal 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 cardinal 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 cardinal number between them.", "questions": [ { "id": "stages-and-transitions-q01", "text": "What identifies and describes the lifecycle of a cardinal 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 cardinal 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 cardinal 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 cardinal number, how often and by whom.", "findings": [ { "id": "observation-record", "name": "Observation record", "description": "How an observation of a cardinal 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 cardinal 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 cardinal 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 cardinal 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 cardinal 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 cardinal 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 cardinal 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 cardinal 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 cardinal 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 cardinal 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 cardinal 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 cardinal number and how confident it is.", "questions": [ { "id": "claim-provenance-q01", "text": "What identifies and describes a claim about a cardinal 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 cardinal 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 cardinal 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 cardinal number is retired without being erased.", "questions": [ { "id": "validity-and-supersession-q01", "text": "What identifies and describes the validity of a claim about a cardinal 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 cardinal 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 cardinal 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 } } } }