axiom
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.
Research draft, second pass
A second pass drafted this model: the structure a model of this thing needs, and what is known about it in the world. The line under this one says how the second half was obtained - researched against sources, or recalled without web access, in which case nothing here was read anywhere and every claim is a lead to verify. Unreviewed either way.
written by Claude from model knowledge without web access - no source was read, every claim is a lead to verify
Researched by: Claude
Purpose and description
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.
A statement accepted as true without proof and taken as a starting point for reasoning, whether a logical axiom valid in all interpretations 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, the fundamental geometric entities and postulates of Euclid, and the closed-world assumption used in logic programming and databases.
What it is for: Founding a formal theory.
It can be explain the concept; relay axiom systems; describe specific axioms; distinguish related concepts.
Distinguishing features
Accepted without proof
Starting point
Logical and non-logical
Consistency matters
What it looks like
Not a visible object; a formal statement.
Physical character
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
How it is recognised
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
Related models
is a kind of - in registry terms
is a kind of - in registry terms
is contrasted with - which is proved
is exemplified by - in set theory
In practice
Families and kinds
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
Standards and regulation
No regulation
Failure modes and hazards
Inconsistent axiom systems such as unrestricted comprehension
Confusing axioms with self-evident truths
Registry aliases mixing axioms with database assumptions
Also called
Where this came from
wikidata · CC0 1.0
Drafted structure
Bundle to layer to finding to question, as the second pass will find it: 4 bundles · 8 layers · 8 findings · 16 questions.
Understand What an axiom is.
Science.
Definition
Definition.
Definition
Definition.
- What is an axiom, and how does it differ from a theorem, definition, postulate and everyday assumption? definition
- Is the question about logic, set theory, geometry, computing or philosophy? boundary
Examples
Examples.
Examples
Examples.
- What are the axiom of empty set, the axiom schema of replacement, unrestricted comprehension and the closed-world assumption? definition
- Which entry fits the specific axiom? action
Logic Logic and set theory.
Science.
Systems
Axiom systems.
Systems
Systems.
- How do ZFC, Peano arithmetic and Hilbert s geometry axiomatise their fields? provenance
- Which references are standard? provenance
Consistency
Consistency and independence.
Consistency
Consistency.
- What do Russell s paradox and Godel s theorems show about axiom systems? provenance
- Which sources are cited? provenance
Philosophy Philosophy.
Attribution.
Status
Status of axioms.
Status
Status.
- Are axioms self-evident, conventional or justified by consequences, with positions attributed? provenance
- Is the presentation attributed? boundary
History
History.
History
History.
- How did the axiomatic method develop from Euclid to Hilbert? provenance
- Which entry fits axiomatic system? action
Context Computing and beyond.
Context.
Computing
Computing.
Computing
Computing.
- How are axioms and assumptions such as the closed-world assumption used in databases and logic programming? provenance
- Which entry fits closed-world assumption? action
Usage
Everyday usage.
Usage
Usage.
- How is axiom used in ordinary language for accepted principles? provenance
- Which entry fits maxim? action
What the second pass must settle
- 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?