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Canonical static concept record

Cardinality of the continuum $\mathfrak c$

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Summary

The size of the real line, denoted $\mathfrak c$ ; in ZFC it is the cardinal $|\mathbb R|=2^{\aleph_0}$ .

Record metadata

Canonical name: Cardinality of the continuum $\mathfrak c$
Concept ID: continuum_cardinal
Content version: 2.3.0
Primary field: Mathematics
Fields: Mathematics
Primary domain: Set theory
Domains: Set theory, Analysis, Number theory
Alternate terminology: No alternate terminology is currently authored for this record.
Concept type: Distinguished cardinal

Carrier(s)

equipotence type of $\mathbb R$ ; an initial ordinal under Choice

Canonically induces

size of $\mathcal P(\mathbb N)$ and of the real line

Notes

In bare ZF, $\mathbb R$ need not be well-orderable, so its size need not have an initial-ordinal representative. The equality with $|\mathcal P(\mathbb N)|$ is equipotence, not literal identity.

Concept sources

Wikipedia — Cardinality of the continuum — Cardinality of the continuum · encyclopedic reference · source ID wp-cardinality-continuum
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.

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

Continuum hypothesis 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: continuum_hypothesis_to_continuum
Relation type: Theorem implication theorem-implication
Direction: source → target
Endpoint roles: source: Implies by theorem; target: Follows by theorem from
Authored annotation: $\mathfrak c=\aleph_1$

Authored explanation

CH identifies the continuum cardinal with the first uncountable cardinal.

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 — Continuum hypothesis — Continuum hypothesis · encyclopedic reference · source ID wp-continuum-hypothesis
Wikipedia — Cardinality of the continuum — Cardinality of the continuum · encyclopedic reference · source ID wp-cardinality-continuum

Power set 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: power_set_to_continuum
Relation type: Theorem implication theorem-implication
Direction: source → target
Endpoint roles: source: Implies by theorem; target: Follows by theorem from
Authored annotation: $|\mathcal P(\mathbb N)|=\mathfrak c$

Authored explanation

Binary sequences, subsets of $\mathbb N$ , and real numbers have the same cardinality.

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 — Cardinality of the continuum — Cardinality of the continuum · encyclopedic reference · source ID wp-cardinality-continuum
Wikipedia — Power set — Power set · encyclopedic reference · source ID wp-power-set

Real numbers $\mathbb R$ 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: real_numbers_to_continuum
Relation type: Theorem implication theorem-implication
Direction: source → target
Endpoint roles: source: Implies by theorem; target: Follows by theorem from
Authored annotation: cardinality $\mathfrak c$

Authored explanation

The real line has cardinality $2^{\aleph_0}$ , denoted $\mathfrak c$ .

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 — Cardinality of the continuum — Cardinality of the continuum · encyclopedic reference · source ID wp-cardinality-continuum
Wikipedia — Construction of the real numbers — Construction of the real numbers · encyclopedic reference · source ID wp-construction-reals

Zermelo–Fraenkel set theory with Choice (ZFC)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: zfc_well_orders_continuum
Relation type: Theorem implication theorem-implication
Direction: source → target
Endpoint roles: source: Implies by theorem; target: Follows by theorem from
Authored annotation: well-orderable continuum

Authored explanation

Choice well-orders $\mathbb R$ , so its equipotence type has a unique initial-ordinal representative $\mathfrak c$ .

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 — Cardinality of the continuum — Cardinality of the continuum · encyclopedic reference · source ID wp-cardinality-continuum
Stanford Encyclopedia of Philosophy — The Axiom of Choice — The Axiom of Choice · scholarly reference · source ID sep-axiom-choice

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Cardinality of the continuum $\mathfrak c$

Summary

Record metadata

Carrier(s)

Canonically induces

Notes

Concept sources

Incoming relations (arrows to this concept)

Beth hierarchy Cardinality of the continuum $\mathfrak c$

Authored explanation

How to interpret this relation type

Relation sources

Continuum hypothesis Cardinality of the continuum $\mathfrak c$

Authored explanation

How to interpret this relation type

Relation sources

Power set Cardinality of the continuum $\mathfrak c$

Authored explanation

How to interpret this relation type

Relation sources

Real numbers $\mathbb R$ Cardinality of the continuum $\mathfrak c$

Authored explanation

How to interpret this relation type

Relation sources

Zermelo–Fraenkel set theory with Choice (ZFC)Cardinality of the continuum $\mathfrak c$

Authored explanation

How to interpret this relation type

Relation sources

Outgoing relations (arrows from this concept)