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Canonical static concept record

Countable infinity $\aleph_0$

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Summary

The least infinite cardinal, represented by the initial ordinal $\omega$ .

Record metadata

Canonical name: Countable infinity $\aleph_0$
Concept ID: aleph_zero
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 cardinal

Carrier(s)

cardinal $\aleph_0$

Canonically induces

cardinality of $\mathbb N$ , $\mathbb Z$ , and $\mathbb Q$

Concept sources

Wikipedia — Aleph number — Aleph number · encyclopedic reference · source ID wp-aleph-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.

Cardinal Countable infinity $\aleph_0$

Permalink to relation

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

Relation ID: cardinal_to_aleph_zero
Relation type: Classification classification
Direction: source → target
Endpoint roles: source: Has subtype; target: Classified as
Authored annotation: least infinite cardinal

Authored explanation

$\aleph_0$ is the first infinite cardinal.

How to interpret this relation type

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

Relation sources

Wikipedia — Aleph number — Aleph number · encyclopedic reference · source ID wp-aleph-number
Open Logic Project — Set Theory — Open Logic Project: Set Theory · open textbook source · source ID openlogic-set-theory

Infinite cardinal Countable infinity $\aleph_0$

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

Relation ID: infinite_cardinal_to_aleph_zero
Relation type: Classification classification
Direction: source → target
Endpoint roles: source: Has subtype; target: Classified as
Authored annotation: least member

Authored explanation

$\aleph_0$ is the least infinite cardinal.

How to interpret this relation type

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

Relation sources

Wikipedia — Aleph number — Aleph number · encyclopedic reference · source ID wp-aleph-number

Integers $\mathbb Z$ Countable infinity $\aleph_0$

Permalink to relation

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

Relation ID: integers_to_aleph_zero
Relation type: Theorem implication theorem-implication
Direction: source → target
Endpoint roles: source: Implies by theorem; target: Follows by theorem from
Authored annotation: countably infinite

Authored explanation

$\mathbb Z$ is countably infinite.

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

nLab — integer — integer · mathematical reference · source ID nlab-integer
Wikipedia — Aleph number — Aleph number · encyclopedic reference · source ID wp-aleph-number

Natural numbers $\mathbb N$ Countable infinity $\aleph_0$

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

Relation ID: natural_numbers_to_aleph_zero
Relation type: Theorem implication theorem-implication
Direction: source → target
Endpoint roles: source: Implies by theorem; target: Follows by theorem from
Authored annotation: cardinality $\aleph_0$

Authored explanation

The carrier of $\mathbb N$ is countably infinite and has cardinality $\aleph_0$ .

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

Wikipedia — Aleph number — Aleph number · encyclopedic reference · source ID wp-aleph-number
nLab — natural number — natural number · mathematical reference · source ID nlab-natural-number

First infinite ordinal $\omega$ Countable infinity $\aleph_0$

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

Relation ID: omega_to_aleph_zero
Relation type: Representation / equivalence representation-equivalence
Direction: source → target
Endpoint roles: source: Representation / equivalence with; target: Representation / equivalence with
Authored annotation: view by cardinality

Authored explanation

The same initial ordinal $\omega$ is denoted $\aleph_0$ when regarded as the least infinite cardinal.

How to interpret this relation type

Reinterpret an object, pass to an equivalent presentation, or relate canonically corresponding structures; the carrier may change.

Relation sources

Wikipedia — Aleph number — Aleph number · encyclopedic reference · source ID wp-aleph-number
nLab — natural number — natural number · mathematical reference · source ID nlab-natural-number

Rational numbers $\mathbb Q$ Countable infinity $\aleph_0$

Permalink to relation

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

Relation ID: rational_numbers_to_aleph_zero
Relation type: Theorem implication theorem-implication
Direction: source → target
Endpoint roles: source: Implies by theorem; target: Follows by theorem from
Authored annotation: countably infinite

Authored explanation

$\mathbb Q$ is countably infinite.

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

nLab — rational number — rational number · mathematical reference · source ID nlab-rational-number
Wikipedia — Aleph number — Aleph number · encyclopedic reference · source ID wp-aleph-number

Outgoing relations (arrows from this concept)

Each relation below starts at this concept.