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Canonical static concept record

First infinite ordinal $\omega$

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Summary

The least infinite ordinal and the set of all finite von Neumann ordinals.

Record metadata

Canonical name: First infinite ordinal $\omega$
Concept ID: omega
Content version: 2.3.0
Primary field: Mathematics
Fields: Mathematics
Primary domain: Set theory
Domains: Set theory, Number theory, Order theory
Alternate terminology: No alternate terminology is currently authored for this record.
Concept type: Distinguished ordinal

Carrier(s)

set $\omega=\{0,1,2,\ldots\}$

Data

membership order

Axioms / constraints

least inductive set

Canonically induces

standard carrier of the natural numbers
first infinite cardinal when viewed by cardinality

Notes

$\omega$ is an ordinal; $\aleph_0$ denotes its role as the least infinite cardinal. In the von Neumann convention they have the same underlying set but different structural roles.

Concept sources

Wikipedia — Von Neumann ordinal — Von Neumann ordinal · encyclopedic reference · source ID wp-von-neumann-ordinal
nLab — natural number — natural number · mathematical reference · source ID nlab-natural-number
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 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

Limit ordinal First infinite ordinal $\omega$

Permalink to relation

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

Relation ID: limit_ordinal_to_omega
Relation type: Classification classification
Direction: source → target
Endpoint roles: source: Has subtype; target: Classified as
Authored annotation: first limit ordinal

Authored explanation

$\omega$ is the least nonzero limit ordinal.

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
Open Logic Project — Set Theory — Open Logic Project: Set Theory · open textbook source · source ID openlogic-set-theory

Ordinal First infinite ordinal $\omega$

Permalink to relation

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

Relation ID: ordinal_to_omega
Relation type: Classification classification
Direction: source → target
Endpoint roles: source: Has subtype; target: Classified as
Authored annotation: first infinite ordinal

Authored explanation

$\omega$ is the first infinite member 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
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.