{"schema":"https://ver.cy/schemas/card/1.0.0","id":"vr.tr.axiom","code":"thing-q17736","url":"https://ver.cy/models/thing/q17736/","name":"axiom","alternateNames":["closed-world assumption","axiom schema of unrestricted comprehension","axiom schema of replacement","axiom of empty set","fundamental geometric entities","logical axiom","axiom of set theory","proper axiom","principle of explosion","Huzita–Hatori axioms","Tarski's axiomatization of the reals","axiom schema of specification"],"kind":"thing","status":"published","version":"0.2.0-wave.4","language":"en","classifiers":{"family":"Thing Registry","category":"Information and virtual systems","entryKind":"thing","plane":"INF","domain":["INF.MED"],"industry":[],"navPath":"","tags":[],"facets":{}},"whatItIs":"A statement accepted as true without proof within a theory and taken as a starting point for reasoning, whether a logical axiom valid in all interpretations of the relevant logic or a non-logical axiom specific to a theory, such as the axioms of set theory including the axiom of empty set, the axiom schema of replacement and the naive axiom schema of unrestricted comprehension that leads to Russell's paradox, or Euclid's geometric postulates; geometric entities are objects described by a theory rather than axioms, and the closed-world assumption used in logic programming and databases is a system assumption rather than a logical axiom.","purpose":"Let an agent explain axioms and their role in logic and mathematics, relay axiom systems and their history from mathematical and philosophical sources, describe the specific axioms and assumptions the registry aliases name, and distinguish axioms from theorems, definitions, postulates in the older sense and assumptions in everyday reasoning.","scope":{"in":[],"out":[],"boundaries":[]},"distinguishingFeatures":["Accepted without proof","Starting point","Logical and non-logical","Consistency matters"],"structure":{"bundles":[{"id":"understand","name":"Understand","description":"What an axiom is.","layers":[{"id":"definition","name":"Definition","description":"Definition.","findings":[{"id":"definition-finding","name":"Definition","description":"Definition.","questions":[{"text":"What is an axiom, and how does it differ from a theorem, definition, postulate and everyday assumption?","kind":"definition"},{"text":"Is the question about logic, set theory, geometry, computing or philosophy?","kind":"boundary"}]}]},{"id":"examples","name":"Examples","description":"Examples.","findings":[{"id":"examples-finding","name":"Examples","description":"Examples.","questions":[{"text":"What are the axiom of empty set, the axiom schema of replacement, unrestricted comprehension and the closed-world assumption?","kind":"definition"},{"text":"Which entry fits the specific axiom?","kind":"action"}]}]}]},{"id":"logic","name":"Logic","description":"Logic and set theory.","layers":[{"id":"systems","name":"Systems","description":"Axiom systems.","findings":[{"id":"systems-finding","name":"Systems","description":"Systems.","questions":[{"text":"How do ZFC, Peano arithmetic and Hilbert s geometry axiomatise their fields?","kind":"provenance"},{"text":"Which references are standard?","kind":"provenance"}]}]},{"id":"consistency","name":"Consistency","description":"Consistency and independence.","findings":[{"id":"consistency-finding","name":"Consistency","description":"Consistency.","questions":[{"text":"What do Russell s paradox and Godel s theorems show about axiom systems?","kind":"provenance"},{"text":"Which sources are cited?","kind":"provenance"}]}]}]},{"id":"philosophy","name":"Philosophy","description":"Philosophy.","layers":[{"id":"status","name":"Status","description":"Status of axioms.","findings":[{"id":"status-finding","name":"Status","description":"Status.","questions":[{"text":"Are axioms self-evident, conventional or justified by consequences, with positions attributed?","kind":"provenance"},{"text":"Is the presentation attributed?","kind":"boundary"}]}]},{"id":"history","name":"History","description":"History.","findings":[{"id":"history-finding","name":"History","description":"History.","questions":[{"text":"How did the axiomatic method develop from Euclid to Hilbert?","kind":"provenance"},{"text":"Which entry fits axiomatic system?","kind":"action"}]}]}]},{"id":"context","name":"Context","description":"Computing and beyond.","layers":[{"id":"computing","name":"Computing","description":"Computing.","findings":[{"id":"computing-finding","name":"Computing","description":"Computing.","questions":[{"text":"How are axioms and assumptions such as the closed-world assumption used in databases and logic programming?","kind":"provenance"},{"text":"Which entry fits closed-world assumption?","kind":"action"}]}]},{"id":"usage","name":"Usage","description":"Everyday usage.","findings":[{"id":"usage-finding","name":"Usage","description":"Usage.","questions":[{"text":"How is axiom used in ordinary language for accepted principles?","kind":"provenance"},{"text":"Which entry fits maxim?","kind":"action"}]}]}]}]},"agentConduct":{"may":["State a theory's axioms and derive consequences from them.","Flag where an axiom set is known to be inconsistent, as unrestricted comprehension is."],"mustNot":["Present an axiom as a self-evident truth rather than an assumption.","Reason from an inconsistent axiom set and present the conclusions as valid.","Switch between axiom systems in one argument without saying so.","Treat a database or system assumption as a logical axiom."],"requiresHuman":["Adopting a different axiom set as the basis of a published proof or a formal system."]},"ethics":{"considerations":["Unstated assumptions let flawed arguments pass as proof, in mathematics and beyond.","Calling a contested assumption axiomatic can shut down legitimate debate."],"affectedParties":["Mathematicians and logicians","Students","Readers of formal arguments"]},"owners":{"steward":"Nobody owns an axiom; the authors of each formal system answer for their choices.","roles":[],"masterSystems":[]},"relations":[{"target":"Q1396667","type":"parent","note":"registry parent class"},{"target":"Q207505","type":"parent","note":"registry parent class"},{"target":"Q536351","type":"parent","note":"registry parent class"},{"target":"postulate","type":"related","note":"in registry terms"},{"target":"first principle","type":"related","note":"in registry terms"},{"target":"theorem","type":"related","note":"which is proved"},{"target":"axiom of choice","type":"related","note":"in set theory"}],"interaction":{"identity":{"applicability":"required","items":["Vercy registry: vr.tr.axiom","Wikidata: Q17736 (https://www.wikidata.org/wiki/Q17736)"]},"properties":{"applicability":"not-applicable","items":["Euclid s Elements: about 300 BCE note - five postulates","ZFC set theory: 1908-1922 note - Zermelo, Fraenkel, Skolem","Russell s paradox: 1901 year - from unrestricted comprehension"]},"recognition":{"applicability":"optional","items":["Unproven starting statement of a theory","Logical axiom, axiom of empty set, axiom schema of replacement, unrestricted comprehension, closed-world assumption, Euclid s postulates","Theorems are proved; definitions introduce terms; assumptions in everyday reasoning are informal","Not a visible object; a formal statement."]},"capabilities":{"applicability":"required","items":["explain the concept","relay axiom systems","describe specific axioms","distinguish related concepts"]},"hazards":{"applicability":"required","items":["Inconsistent axiom systems such as unrestricted comprehension","Confusing axioms with self-evident truths","Registry aliases mixing axioms with database assumptions"]},"interfaces":{"applicability":"required","items":["No regulation"]},"context":{"applicability":"required","items":["Founding a formal theory.","logical axioms","axioms of set theory such as empty set and replacement","axioms of geometry and arithmetic","axioms of algebraic structures","assumptions in computing such as the closed-world assumption","historical postulates"]}},"sources":[{"title":"Wikidata item Q17736: axiom","url":"https://www.wikidata.org/wiki/Q17736","note":"identity and sense of the item"},{"title":"Wikipedia: Axiom","url":"https://en.wikipedia.org/wiki/Axiom","note":"general description of the item"}],"openQuestions":["Should the set theory axioms and the closed-world assumption be separate primary entries?","How should mathematical sources be linked?","The registry entry has merged aliases naming specific axioms and a computing assumption; should they be split off?"],"resources":{"spec":"/models/things/publications/thing-q17736/spec.json"},"provenance":{"origin":"thing registry research (pass 2)","builtFrom":["models/things/publications/thing-q17736/spec.json"],"providers":["Claude"],"researchStatus":"unreviewed","generatedAt":"2026-09-13T14:49:35Z","builder":"tools/build_cards.py@1.0.0"},"completeness":{"sections":{"classifiers":"filled","whatItIs":"filled","purpose":"filled","distinguishingFeatures":"filled","structure":"filled","agentConduct":"filled","ethics":"filled","owners":"filled","relations":"filled","interaction.identity":"filled","interaction.properties":"not-applicable","interaction.recognition":"filled","interaction.capabilities":"filled","interaction.hazards":"filled","interaction.interfaces":"filled","interaction.context":"filled","sources":"filled"},"notes":{"_gate":"Published through the thing publication gate 1.0 on 2026-10-07T10:10:26+00:00: completeness 1.00; sense: Wikidata description; 2 live sources; review by codex: pass","interaction.properties":"Plane INF: no invented physical properties."},"score":1.0}}