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Canonical static concept record

Finite ordinal

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Summary

A von Neumann ordinal $n$ obtained from $0=\varnothing$ by finitely many successor steps, so $n=\{0,1,\ldots,n-1\}$ .

Record metadata

Canonical name: Finite ordinal
Concept ID: finite_ordinal
Content version: 2.3.0
Primary field: Mathematics
Fields: Mathematics
Primary domain: Set theory
Domains: Set theory, Number theory
Alternate terminology: No alternate terminology is currently authored for this record.
Concept type: Ordinal class

Carrier(s)

transitive finite set $n$

Axioms / constraints

membership well-orders $n$

Canonically induces

finite cardinal and natural-number numeral

Notes

As examples, $0=\varnothing$ , $1=\{0\}$ , $2=\{0,1\}$ , $3=\{0,1,2\}$ , and so on, or in terms of empty sets: $0=\varnothing$ , $1=\{\varnothing\}$ , $2=\{\varnothing,\{\varnothing\}\}$ , $3=\{\varnothing,\{\varnothing\},\{\varnothing,\{\varnothing\}\}\}$ , and so on.Finite ordinals are canonical representatives of finite order types and finite cardinalities.

Concept sources

Wikipedia — Von Neumann ordinal — Von Neumann ordinal · encyclopedic reference · source ID wp-von-neumann-ordinal
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.

Finite set Finite ordinal

Permalink to relation

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

Relation ID: finite_set_to_finite_ordinal
Relation type: Impose axiom impose-axiom
Direction: source → target
Endpoint roles: source: Builds toward; target: Built from
Authored annotation: require ordinal form

Authored explanation

Select the finite sets that are transitive and well-ordered by membership; these are the finite von Neumann ordinals.

How to interpret this relation type

Keep the existing data and select the subclass satisfying an additional law, existence condition, finiteness condition, or other property.

Relation sources

Wikipedia — Von Neumann ordinal — Von Neumann ordinal · encyclopedic reference · source ID wp-von-neumann-ordinal
Open Logic Project — Set Theory — Open Logic Project: Set Theory · open textbook source · source ID openlogic-set-theory

Ordinal Finite ordinal

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

Relation ID: ordinal_to_finite_ordinal
Relation type: Classification classification
Direction: source → target
Endpoint roles: source: Has subtype; target: Classified as
Authored annotation: finite ordinals

Authored explanation

Finite ordinals are the finite members of the ordinal hierarchy.

How to interpret this relation type

The target is a member or subtype of the broader source class.

Relation sources

Wikipedia — Ordinal number — Ordinal number · expository · source ID wp-ordinal
Wikipedia — Von Neumann ordinal — Von Neumann ordinal · encyclopedic reference · source ID wp-von-neumann-ordinal

Von Neumann successor Finite ordinal

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

Relation ID: von_neumann_successor_to_finite_ordinal
Relation type: Canonical construction canonical-construction
Direction: source → target
Endpoint roles: source: Canonically constructs; target: Canonically constructed from
Authored annotation: iterate from zero

Authored explanation

Finite iteration of von Neumann successor from $0$ produces exactly the finite ordinals.

How to interpret this relation type

Apply a standard functorial or canonical construction whose output is not merely a reduct of the input and is not generally an equivalent presentation of the same object.

Relation sources

Wikipedia — Von Neumann ordinal — Von Neumann ordinal · encyclopedic reference · source ID wp-von-neumann-ordinal
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.

Finite ordinal First infinite ordinal $\omega$

Permalink to relation

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

Relation ID: finite_ordinal_to_omega
Relation type: Canonical construction canonical-construction
Direction: source → target
Endpoint roles: source: Canonically constructs; target: Canonically constructed from
Authored annotation: collect all finite ordinals

Authored explanation

$\omega$ is the set of all finite ordinals.

How to interpret this relation type

Apply a standard functorial or canonical construction whose output is not merely a reduct of the input and is not generally an equivalent presentation of the same object.

Relation sources

Wikipedia — Von Neumann ordinal — Von Neumann ordinal · encyclopedic reference · source ID wp-von-neumann-ordinal
Open Logic Project — Set Theory — Open Logic Project: Set Theory · open textbook source · source ID openlogic-set-theory

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Finite ordinal

Summary

Record metadata

Carrier(s)

Axioms / constraints

Canonically induces

Notes

Concept sources

Incoming relations (arrows to this concept)

Finite set Finite ordinal

Authored explanation

How to interpret this relation type

Relation sources

Ordinal Finite ordinal

Authored explanation

How to interpret this relation type

Relation sources

Von Neumann successor Finite ordinal

Authored explanation

How to interpret this relation type

Relation sources

Outgoing relations (arrows from this concept)

Finite ordinal First infinite ordinal $\omega$

Authored explanation

How to interpret this relation type

Relation sources