{ "model": { "rank": 6781, "code": "thing-q4049983", "model_id": "vr.tr.cardinality", "name": "cardinality", "purpose": "Let an agent explain cardinality in set theory and its applications, compare finite and infinite cardinalities, and separate the database and modelling senses.", "family": "Thing Registry", "category": "Cross-cutting context", "status": "research-draft", "kind": "thing", "plane": "XCT", "domain": "XCT.QLT", "industry": "", "version": "", "url": "/models/thing/q4049983/", "tier": 2, "score": 58, "payload": { "layer": "wikidata", "aliases": [ "uniqueness", "group order", "number of entities", "order type", "independence number", "clique number", "vertex cover number", "matching number", "edge cover number", "class number", "number of prime divisors", "number of atoms in the unit cell", "number of possible states of a storage device", "number of molecules", "number of base pairs", "number of quarks", "number of antiquarks", "number of comments", "number of likes", "number of dislikes", "number of reblogs", "number of viewers/listeners", "number of subscribers", "passenger count", "pulse count", "number of works", "number of liked posts", "number of followed accounts", "stroke count", "number of configurations", "number of rooms", "number of pages", "number of enemies", "edition count", "degree of comparison", "trunk count", "max leaf number", "statistics in Pokémon", "lateral line scale count", "number of gill rakers" ], "aliasCount": 63, "merged": 63, "knownIn": 58, "facets": null, "markers": [], "lexicalClass": "", "senseRank": null, "alsoRegisteredAs": "vr.tr.cardinality", "source": { "dataset": "wikidata", "item": "Q4049983", "url": "https://www.wikidata.org/wiki/Q4049983", "license": "CC0 1.0" } }, "research": { "vercy": "1.0-draft", "publication": { "status": "research-draft", "adjudicationStatus": "unreviewed", "publishableCanonical": false, "generatedAt": "2026-09-11T15:46:30Z", "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": 1, "engine": "claude" }, "metaModel": { "id": "THING-Q4049983", "registryId": "vr.tr.cardinality", "name": "cardinality", "version": "0.2.0-wave.1", "entryKind": "thing", "family": "Thing Registry", "domain": [ "XCT.QLT" ], "status": "research-draft" }, "canonicalUrl": "https://ver.cy/models/thing/q4049983/", "model": { "registry_id": "vr.tr.cardinality", "name": "cardinality", "purpose": "Let an agent explain cardinality in set theory and its applications, compare finite and infinite cardinalities, and separate the database and modelling senses.", "definition": "The number of elements in a set, generalised to infinite sets by Cantor, where sets have the same cardinality if there is a one-to-one correspondence between them; related notions include the order of a group, the clique and independence numbers of graphs, and in databases and modelling the number of related entities or distinct values.", "what_it_is_for": "Measuring the size of sets and collections.", "affordances": [ "determine the cardinality of a set", "explain countable and uncountable infinities", "apply cardinality in graphs and groups", "explain cardinality in data modelling" ], "distinguishing_features": [ "Size of a set", "Defined by bijection", "Finite and infinite", "Applied widely" ], "appearance": "Not physical; a mathematical quantity.", "visual_identification": [ "Vertical bars notation for set size", "Aleph numbers for infinite sets", "Ordinal numbers describe order, not size" ], "physical_properties": [], "families_and_kinds": [ "finite cardinalities", "countable infinity", "uncountable cardinalities", "graph and group invariants", "cardinality in databases" ], "related_models": [ { "relation": "is a kind of", "target": "cardinal function", "why": "category" }, { "relation": "is a kind of", "target": "index number", "why": "category" }, { "relation": "is related to", "target": "binomial coefficient", "why": "counting" }, { "relation": "is related to", "target": "probability distribution", "why": "sample space size" } ], "identifiers": [], "standards_and_regulation": [ "ISO 80000-2 notation" ], "failure_modes_and_hazards": [ "Confusing cardinality with ordinality", "Assuming all infinities are equal", "Confusing database and set senses" ], "in_scope": [], "out_of_scope": [], "characteristics": [] }, "sources": [], "structure": { "bundles": [ { "id": "finite", "name": "Finite", "description": "Counting elements.", "rationale": "Count carefully.", "layers": [ { "id": "count", "name": "Count", "description": "Size of a set.", "findings": [ { "id": "count-finding", "name": "Count", "description": "Size.", "questions": [ { "text": "What is the cardinality of this set?", "kind": "measurement" }, { "text": "Are duplicates excluded?", "kind": "boundary" } ] } ] }, { "id": "combine", "name": "Combine", "description": "Unions and products.", "findings": [ { "id": "combine-finding", "name": "Combine", "description": "Combinations.", "questions": [ { "text": "What is the cardinality of the union, intersection or product of these sets?", "kind": "measurement" }, { "text": "Which counting principle applies?", "kind": "definition" } ] } ] } ] }, { "id": "infinite", "name": "Infinite", "description": "Infinities.", "rationale": "Not all infinities are equal.", "layers": [ { "id": "countable", "name": "Countable", "description": "Countable sets.", "findings": [ { "id": "countable-finding", "name": "Countable", "description": "Countable sets.", "questions": [ { "text": "Is this set countable, and how is a bijection with the natural numbers shown?", "kind": "definition" }, { "text": "Why are the rationals countable?", "kind": "definition" } ] } ] }, { "id": "uncountable", "name": "Uncountable", "description": "Larger infinities.", "findings": [ { "id": "uncountable-finding", "name": "Uncountable", "description": "Uncountable sets.", "questions": [ { "text": "Why are the real numbers uncountable, according to the diagonal argument?", "kind": "definition" }, { "text": "What is the continuum hypothesis?", "kind": "provenance" } ] } ] } ] }, { "id": "applied", "name": "Applied", "description": "Other uses.", "rationale": "Same idea, other fields.", "layers": [ { "id": "graphs", "name": "Graphs", "description": "Graph invariants.", "findings": [ { "id": "graphs-finding", "name": "Graphs", "description": "Graph invariants.", "questions": [ { "text": "What are the clique number and independence number of this graph?", "kind": "measurement" }, { "text": "How hard are they to compute?", "kind": "provenance" } ] } ] }, { "id": "groups", "name": "Groups", "description": "Group order.", "findings": [ { "id": "groups-finding", "name": "Groups", "description": "Group order.", "questions": [ { "text": "What is the order of this group?", "kind": "measurement" }, { "text": "What does Lagrange theorem say about subgroup orders?", "kind": "definition" } ] } ] } ] }, { "id": "data", "name": "Data", "description": "Modelling sense.", "rationale": "Different meaning.", "layers": [ { "id": "database", "name": "Database", "description": "Relationship cardinality.", "findings": [ { "id": "database-finding", "name": "Database", "description": "Database cardinality.", "questions": [ { "text": "Is cardinality meant as one-to-many relationships or distinct value counts in a database?", "kind": "boundary" }, { "text": "How is it modelled?", "kind": "definition" } ] } ] }, { "id": "teach", "name": "Teach", "description": "Teaching.", "findings": [ { "id": "teach-finding", "name": "Teach", "description": "Teaching.", "questions": [ { "text": "How can infinite cardinality be taught intuitively?", "kind": "action" }, { "text": "Which misconceptions arise?", "kind": "provenance" } ] } ] } ] } ] }, "openQuestions": [ "Should infinite cardinals be a separate entry?", "How should graph invariants be linked?", "How should the database sense be split?" ], "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 cardinality, record what state it is in, and decide what may be done with it.", "whatItIs": "Let an agent explain cardinality in set theory and its applications, compare finite and infinite cardinalities, and separate the database and modelling senses.", "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: uniqueness, group order, number of entities, order type, independence number, clique number, vertex cover number, matching number, edge cover number, class number, number of prime divisors, number of atoms in the unit cell.", "63 finer distinctions are held as aliases rather than separate entries, because telling them apart needs a task that asks for it.", "Described in 58 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 cardinality, 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": [ "uniqueness", "group order", "number of entities", "order type", "independence number", "clique number", "vertex cover number", "matching number", "edge cover number", "class number", "number of prime divisors", "number of atoms in the unit cell" ] } ], "standing": "Described in 58 Wikipedia languages, which measures how widely it is written about rather than how important or how common it is. 63 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 cardinality from another, and a cardinality 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 cardinality, 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 cardinality, 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 cardinality 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 cardinality, 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 cardinality, 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 cardinality 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 cardinality 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 cardinality, 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 cardinality, 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 cardinality 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 cardinality from the things most often confused with it.", "questions": [ { "id": "distinguishing-features-q01", "text": "What identifies and describes what distinguishes a cardinality, 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 cardinality, 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 cardinality 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 cardinality 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 cardinality between them.", "questions": [ { "id": "stages-and-transitions-q01", "text": "What identifies and describes the lifecycle of a cardinality, 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 cardinality, 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 cardinality 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 cardinality, how often and by whom.", "findings": [ { "id": "observation-record", "name": "Observation record", "description": "How an observation of a cardinality 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 cardinality, 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 cardinality, 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 cardinality 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 cardinality 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 cardinality 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 cardinality 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 cardinality 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 cardinality?", "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 cardinality?", "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 cardinality 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 cardinality and how confident it is.", "questions": [ { "id": "claim-provenance-q01", "text": "What identifies and describes a claim about a cardinality, 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 cardinality, 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 cardinality 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 cardinality is retired without being erased.", "questions": [ { "id": "validity-and-supersession-q01", "text": "What identifies and describes the validity of a claim about a cardinality, 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 cardinality, 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 cardinality 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 } } } }