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Canonical static concept record

Beth hierarchy

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Summary

In ZFC, the cardinal hierarchy generated from $\beth_0=\aleph_0$ by repeated power sets and suprema at limit stages.

Record metadata

Canonical name: Beth hierarchy
Concept ID: beth_hierarchy
Content version: 2.3.0
Primary field: Mathematics
Fields: Mathematics
Primary domain: Set theory
Domains: Set theory, Logic
Alternate terminology: No alternate terminology is currently authored for this record.
Concept type: Cardinal hierarchy

Carrier(s)

class-indexed family of cardinals

Data

$\beth_{\alpha+1}=2^{\beth_\alpha}$
$\beth_\lambda=\sup_{\alpha<\lambda}\beth_\alpha$

Notes

Without Choice, a power set of a well-orderable set need not be well-orderable; the displayed initial-ordinal beth hierarchy therefore uses the ZFC setting selected later in the story.

Concept sources

Wikipedia — Beth number — Beth number · encyclopedic reference · source ID wp-beth-number
Wikipedia — Cantor’s theorem — Cantor’s theorem · encyclopedic reference · source ID wp-cantor-theorem

Incoming relations (arrows to this concept)

Each relation below ends at this concept.

Countable infinity $\aleph_0$ Beth hierarchy

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

Relation ID: aleph_zero_to_beth_hierarchy
Relation type: Canonical construction canonical-construction
Direction: source → target
Endpoint roles: source: Canonically constructs; target: Canonically constructed from
Authored annotation: start at $\beth_0$

Authored explanation

Set $\beth_0=\aleph_0$ and iterate cardinal exponentiation by $2$ .

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 — Beth number — Beth number · encyclopedic reference · source ID wp-beth-number

Cantor’s theorem Beth hierarchy

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

Relation ID: cantors_theorem_to_beth_hierarchy
Relation type: Theorem implication theorem-implication
Direction: source → target
Endpoint roles: source: Implies by theorem; target: Follows by theorem from
Authored annotation: strict growth at successors

Authored explanation

Cantor’s theorem ensures $\beth_\alpha<\beth_{\alpha+1}$ .

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 — Cantor’s theorem — Cantor’s theorem · encyclopedic reference · source ID wp-cantor-theorem
Wikipedia — Beth number — Beth number · encyclopedic reference · source ID wp-beth-number

Generalized continuum hypothesis Beth hierarchy

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

Relation ID: gch_to_aleph_beth
Relation type: Theorem implication theorem-implication
Direction: source → target
Endpoint roles: source: Implies by theorem; target: Follows by theorem from
Authored annotation: $\beth_\alpha=\aleph_\alpha$

Authored explanation

GCH identifies the beth and aleph hierarchies at every ordinal index.

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 — Beth number — Beth number · encyclopedic reference · source ID wp-beth-number
Wikipedia — Continuum hypothesis — Continuum hypothesis · encyclopedic reference · source ID wp-continuum-hypothesis

Power set Beth hierarchy

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

Relation ID: power_set_to_beth_hierarchy
Relation type: Canonical construction canonical-construction
Direction: source → target
Endpoint roles: source: Canonically constructs; target: Canonically constructed from
Authored annotation: power-set successor stages

Authored explanation

In ZFC, the step $\beth_{\alpha+1}=2^{\beth_\alpha}$ records the initial-ordinal cardinality of a power set.

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 — Beth number — Beth number · encyclopedic reference · source ID wp-beth-number
Wikipedia — Power set — Power set · encyclopedic reference · source ID wp-power-set

Zermelo–Fraenkel set theory with Choice (ZFC)Beth hierarchy

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

Relation ID: zfc_supports_beth_hierarchy
Relation type: Theorem implication theorem-implication
Direction: source → target
Endpoint roles: source: Implies by theorem; target: Follows by theorem from
Authored annotation: well-order power-set sizes

Authored explanation

Choice ensures that every power set appearing in the beth recursion has an initial-ordinal cardinal representative.

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 — Axiom of choice — Axiom of choice · encyclopedic reference · source ID wp-axiom-choice
Wikipedia — Beth number — Beth number · encyclopedic reference · source ID wp-beth-number
Stanford Encyclopedia of Philosophy — The Axiom of Choice — The Axiom of Choice · scholarly reference · source ID sep-axiom-choice

Outgoing relations (arrows from this concept)

Each relation below starts at this concept.

Beth hierarchy Cardinality of the continuum $\mathfrak c$

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

Relation ID: beth_hierarchy_to_continuum
Relation type: Classification classification
Direction: source → target
Endpoint roles: source: Has subtype; target: Classified as
Authored annotation: first power-set step

Authored explanation

$\mathfrak c=\beth_1=2^{\aleph_0}$ .

How to interpret this relation type

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

Relation sources

Wikipedia — Beth number — Beth number · encyclopedic reference · source ID wp-beth-number
Wikipedia — Cardinality of the continuum — Cardinality of the continuum · encyclopedic reference · source ID wp-cardinality-continuum

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Beth hierarchy

Summary

Record metadata

Carrier(s)

Data

Notes

Concept sources

Incoming relations (arrows to this concept)

Countable infinity $\aleph_0$ Beth hierarchy

Authored explanation

How to interpret this relation type

Relation sources

Cantor’s theorem Beth hierarchy

Authored explanation

How to interpret this relation type

Relation sources

Generalized continuum hypothesis Beth hierarchy

Authored explanation

How to interpret this relation type

Relation sources

Power set Beth hierarchy

Authored explanation

How to interpret this relation type

Relation sources

Zermelo–Fraenkel set theory with Choice (ZFC)Beth hierarchy

Authored explanation

How to interpret this relation type

Relation sources

Outgoing relations (arrows from this concept)

Beth hierarchy Cardinality of the continuum $\mathfrak c$

Authored explanation

How to interpret this relation type

Relation sources