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Canonical static concept record

First uncountable ordinal $\omega_1$

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Summary

The least uncountable ordinal, equivalently the set of all countable ordinals.

Record metadata

Canonical name: First uncountable ordinal $\omega_1$
Concept ID: first_uncountable_ordinal
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: Distinguished ordinal

Carrier(s)

ordinal $\omega_1$

Axioms / constraints

every $\alpha<\omega_1$ is countable

Canonically induces

initial ordinal $\aleph_1$

Notes

Its underlying cardinal is $\aleph_1$ ; the continuum hypothesis asks whether $|\mathbb R|=\aleph_1$ .

Concept sources

Wikipedia — First uncountable ordinal — First uncountable ordinal · encyclopedic reference · source ID wp-first-uncountable-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.

Countable ordinal First uncountable ordinal $\omega_1$

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

Relation ID: countable_ordinal_to_omega1
Relation type: Canonical construction canonical-construction
Direction: source → target
Endpoint roles: source: Canonically constructs; target: Canonically constructed from
Authored annotation: collect all countable ordinals

Authored explanation

$\omega_1$ is the set of all countable 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 — First uncountable ordinal — First uncountable ordinal · encyclopedic reference · source ID wp-first-uncountable-ordinal
Open Logic Project — Set Theory — Open Logic Project: Set Theory · open textbook source · source ID openlogic-set-theory

Ordinal First uncountable ordinal $\omega_1$

Permalink to relation

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

Relation ID: ordinal_to_omega1
Relation type: Classification classification
Direction: source → target
Endpoint roles: source: Has subtype; target: Classified as
Authored annotation: first uncountable ordinal

Authored explanation

$\omega_1$ is the least uncountable 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 — First uncountable ordinal — First uncountable ordinal · encyclopedic reference · source ID wp-first-uncountable-ordinal

Outgoing relations (arrows from this concept)

Each relation below starts at this concept.

No direct relations are authored in this direction.

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First uncountable ordinal $\omega_1$

Summary

Record metadata

Carrier(s)

Axioms / constraints

Canonically induces

Notes

Concept sources

Incoming relations (arrows to this concept)

Countable ordinal First uncountable ordinal $\omega_1$

Authored explanation

How to interpret this relation type

Relation sources

Ordinal First uncountable ordinal $\omega_1$

Authored explanation

How to interpret this relation type

Relation sources

Outgoing relations (arrows from this concept)