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axiom

vr.tr.axiom · thing-q17736

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.

Thing Registry Information and virtual systems INF.MED
Bundle → Layer → Finding → QuestionsClassifiersWhat it isWhy it existsDistinguishing featuresWhat robots and AI may and may not doMoral aspectsOwnersLinks to other meta-modelsIdentity and identifiersDirect propertiesRecognitionCapabilities and actionsHazards and failure modesStandards and interfacesContext of useSources

Bundle → Layer → Finding → Questions Filled

4 bundles · 8 layers · 8 findings · 16 questions

Understand What an axiom is.

Definition

Definition.

Definition

Definition.

  1. What is an axiom, and how does it differ from a theorem, definition, postulate and everyday assumption? definition
  2. Is the question about logic, set theory, geometry, computing or philosophy? boundary

Examples

Examples.

Examples

Examples.

  1. What are the axiom of empty set, the axiom schema of replacement, unrestricted comprehension and the closed-world assumption? definition
  2. Which entry fits the specific axiom? action
Logic Logic and set theory.

Systems

Axiom systems.

Systems

Systems.

  1. How do ZFC, Peano arithmetic and Hilbert s geometry axiomatise their fields? provenance
  2. Which references are standard? provenance

Consistency

Consistency and independence.

Consistency

Consistency.

  1. What do Russell s paradox and Godel s theorems show about axiom systems? provenance
  2. Which sources are cited? provenance
Philosophy Philosophy.

Status

Status of axioms.

Status

Status.

  1. Are axioms self-evident, conventional or justified by consequences, with positions attributed? provenance
  2. Is the presentation attributed? boundary

History

History.

History

History.

  1. How did the axiomatic method develop from Euclid to Hilbert? provenance
  2. Which entry fits axiomatic system? action
Context Computing and beyond.

Computing

Computing.

Computing

Computing.

  1. How are axioms and assumptions such as the closed-world assumption used in databases and logic programming? provenance
  2. Which entry fits closed-world assumption? action

Usage

Everyday usage.

Usage

Usage.

  1. How is axiom used in ordinary language for accepted principles? provenance
  2. Which entry fits maxim? action

Classifiers Filled

Family
Thing Registry
Category
Information and virtual systems
Entry kind
thing
Plane
INF
Domain
INF.MED
Other names and narrower kinds
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

What it is Filled

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.

Why it exists Filled

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.

Distinguishing features Filled

  • Accepted without proof
  • Starting point
  • Logical and non-logical
  • Consistency matters

What robots and AI may and may not do Filled

Must not

  • 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.

Only with a human decision

  • Adopting a different axiom set as the basis of a published proof or a formal system.

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.

Moral aspects Filled

  • Unstated assumptions let flawed arguments pass as proof, in mathematics and beyond.
  • Calling a contested assumption axiomatic can shut down legitimate debate.

Who is affected

  • Mathematicians and logicians
  • Students
  • Readers of formal arguments

Owners Filled

Steward

Nobody owns an axiom; the authors of each formal system answer for their choices.

Links to other meta-models Filled

parent

  • Q1396667 - registry parent class
  • Q207505 - registry parent class
  • Q536351 - registry parent class

related

  • postulate - in registry terms
  • first principle - in registry terms
  • theorem - which is proved
  • axiom of choice - in set theory

What else AI and robots need to interact with it Filled

Identity and identifiers required Filled

  • Vercy registry: vr.tr.axiom
  • Wikidata: Q17736 (https://www.wikidata.org/wiki/Q17736)

Direct properties not applicable Not applicable

  • 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

Plane INF: no invented physical properties.

Recognition optional Filled

  • 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 and actions required Filled

  • explain the concept
  • relay axiom systems
  • describe specific axioms
  • distinguish related concepts

Hazards and failure modes required Filled

  • Inconsistent axiom systems such as unrestricted comprehension
  • Confusing axioms with self-evident truths
  • Registry aliases mixing axioms with database assumptions

Standards and interfaces required Filled

  • No regulation

Context of use required Filled

  • 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 Filled

  1. Wikidata item Q17736: axiom - identity and sense of the item
  2. Wikipedia: Axiom - general description of the item

Open questions

  • 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?

Machine files

Card as JSON Specification Card backlog vercy.card/1

Provenance

thing registry research (pass 2) · unreviewed

Built from: models/things/publications/thing-q17736/spec.json