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

Epsilon-zero ordinal $\varepsilon_0$

Open this concept in the interactive graph Read the Markdown equivalent

Summary

The least nonzero fixed point of ordinal exponentiation $\alpha\mapsto\omega^\alpha$ .

Record metadata

Canonical name: Epsilon-zero ordinal $\varepsilon_0$
Concept ID: epsilon_zero
Content version: 2.3.0
Primary field: Mathematics
Fields: Mathematics
Primary domain: Set theory
Domains: Set theory, Logic, Order theory
Alternate terminology: No alternate terminology is currently authored for this record.
Concept type: Distinguished ordinal

Carrier(s)

ordinal $\varepsilon_0$

Axioms / constraints

$\omega^{\varepsilon_0}=\varepsilon_0$

Canonically induces

proof-theoretic ordinal of Peano arithmetic

Notes

$\varepsilon_0$ is countable despite lying far beyond every finite tower formed from $\omega$ by ordinal exponentiation.

Concept sources

Wikipedia — Epsilon number — Epsilon number · encyclopedic reference · source ID wp-epsilon-number
Wikipedia — Ordinal arithmetic — Ordinal arithmetic · encyclopedic reference · source ID wp-ordinal-arithmetic

Incoming relations (arrows to this concept)

Each relation below ends at this concept.

Countable ordinal Epsilon-zero ordinal $\varepsilon_0$

Permalink to relation

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

Relation ID: countable_ordinal_to_epsilon_zero
Relation type: Classification classification
Direction: source → target
Endpoint roles: source: Has subtype; target: Classified as
Authored annotation: distinguished countable ordinal

Authored explanation

$\varepsilon_0$ is a countable ordinal.

How to interpret this relation type

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

Relation sources

Wikipedia — Epsilon number — Epsilon number · encyclopedic reference · source ID wp-epsilon-number

Ordinal arithmetic Epsilon-zero ordinal $\varepsilon_0$

Permalink to relation

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

Relation ID: ordinal_arithmetic_to_epsilon_zero
Relation type: Canonical construction canonical-construction
Direction: source → target
Endpoint roles: source: Canonically constructs; target: Canonically constructed from
Authored annotation: least fixed point of $\omega^\alpha$

Authored explanation

Iterating $\alpha\mapsto\omega^\alpha$ from finite stages and taking the supremum yields $\varepsilon_0$ .

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 — Epsilon number — Epsilon number · encyclopedic reference · source ID wp-epsilon-number
Wikipedia — Ordinal arithmetic — Ordinal arithmetic · encyclopedic reference · source ID wp-ordinal-arithmetic

Outgoing relations (arrows from this concept)

Each relation below starts at this concept.

No direct relations are authored in this direction.

Atlas of Fundamental Concepts

Select a concept

Move selected concept

Stories & Views

Compare concepts

Guide to the Atlas

Epsilon-zero ordinal $\varepsilon_0$

Summary

Record metadata

Carrier(s)

Axioms / constraints

Canonically induces

Notes

Concept sources

Incoming relations (arrows to this concept)

Countable ordinal Epsilon-zero ordinal $\varepsilon_0$

Authored explanation

How to interpret this relation type

Relation sources

Ordinal arithmetic Epsilon-zero ordinal $\varepsilon_0$

Authored explanation

How to interpret this relation type

Relation sources

Outgoing relations (arrows from this concept)