Select any concept, construction junction, or annotated relation by pointer, touch, search, or keyboard.

Construction junctions are diamonds. They show where multiple structures must coexist on the same carrier and satisfy compatibility conditions.

Canonical static concept record

Well-ordering theorem

Open this concept in the interactive graph Read the Markdown equivalent

Summary

Every set can be well-ordered.

Record metadata

Canonical name: Well-ordering theorem
Concept ID: well_ordering_theorem
Content version: 2.3.0
Primary field: Mathematics
Fields: Mathematics
Primary domain: Set theory
Domains: Set theory, Order theory, Logic
Alternate terminology: No alternate terminology is currently authored for this record.
Concept type: Theorem

Carrier(s)

set-theoretic statement

Canonically induces

a cardinal representative for every set

Notes

Over ZF, the well-ordering theorem is equivalent to the axiom of choice.

Concept sources

Wikipedia — Well-ordering theorem — Well-ordering theorem · encyclopedic reference · source ID wp-well-ordering-theorem
nLab — Well-ordering theorem — Well-ordering theorem · mathematical reference · source ID nlab-well-ordering-theorem
Stanford Encyclopedia of Philosophy — The Axiom of Choice — The Axiom of Choice · scholarly reference · source ID sep-axiom-choice
Open Logic Project — Set Theory — Open Logic Project: Set Theory · open textbook source · source ID openlogic-set-theory

Incoming relations (arrows to this concept)

Each relation below ends at this concept.

Axiom of choice (AC)Well-ordering theorem

Permalink to relation

This is an authored directed relation from the source endpoint to the target endpoint.

Relation ID: choice_to_well_ordering
Relation type: Theorem implication theorem-implication
Direction: source → target
Endpoint roles: source: Implies by theorem; target: Follows by theorem from
Authored annotation: equivalent over ZF

Authored explanation

Over ZF, AC implies that every set can be well-ordered; conversely, the well-ordering theorem implies AC.

How to interpret this relation type

Record a genuine theorem implication that is not part of the target definition; these edges may point toward a weaker structure.

Relation sources

Stanford Encyclopedia of Philosophy — The Axiom of Choice — The Axiom of Choice · scholarly reference · source ID sep-axiom-choice
Open Logic Project — Set Theory — Open Logic Project: Set Theory · open textbook source · source ID openlogic-set-theory

Outgoing relations (arrows from this concept)

Each relation below starts at this concept.