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Canonical static concept record

Power set

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Summary

For a set $X$ , the set $\mathcal P(X)$ of all subsets of $X$ .

Record metadata

Canonical name: Power set
Concept ID: power_set
Content version: 2.3.0
Primary field: Mathematics
Fields: Mathematics
Primary domain: Set theory
Domains: Set theory
Alternate terminology: No alternate terminology is currently authored for this record.
Concept type: Canonical construction

Carrier(s)

set $X$

Data

$\mathcal P(X)=\{A:A\subseteq X\}$

Canonically induces

strictly larger cardinality by Cantor’s theorem

Notes

In ZF, the power-set axiom guarantees that $\mathcal P(X)$ is a set.

Concept sources

Wikipedia — Power set — Power set · encyclopedic reference · source ID wp-power-set
nLab — Power set — Power set · mathematical reference · source ID nlab-power-set
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.

Set Power set

Permalink to relation

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

Relation ID: set_to_power_set
Relation type: Canonical construction canonical-construction
Direction: source → target
Endpoint roles: source: Canonically constructs; target: Canonically constructed from
Authored annotation: form all subsets

Authored explanation

Send $X$ to $\mathcal P(X)$ , the set of all subsets of $X$ .

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 — Power set — Power set · encyclopedic reference · source ID wp-power-set
Open Logic Project — Set Theory — Open Logic Project: Set Theory · open textbook source · source ID openlogic-set-theory

Zermelo–Fraenkel set theory (ZF)Power set

Permalink to relation

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

Relation ID: zf_provides_power_sets
Relation type: Theorem implication theorem-implication
Direction: source → target
Endpoint roles: source: Implies by theorem; target: Follows by theorem from
Authored annotation: power-set axiom

Authored explanation

The power-set axiom of ZF asserts that for every set $X$ , the collection $\mathcal P(X)$ is a set.

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 — Zermelo–Fraenkel set theory — Zermelo–Fraenkel set theory · encyclopedic reference · source ID wp-zf
Wikipedia — Power set — Power set · encyclopedic reference · source ID wp-power-set
Open Logic Project — Set Theory — Open Logic Project: Set Theory · open textbook source · source ID openlogic-set-theory

Outgoing relations (arrows from this concept)

Each relation below starts at this concept.

Power set Beth hierarchy

Permalink to relation

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

Power set Cantor’s theorem

Permalink to relation

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

Relation ID: power_set_to_cantors_theorem
Relation type: Theorem implication theorem-implication
Direction: source → target
Endpoint roles: source: Implies by theorem; target: Follows by theorem from
Authored annotation: strictly larger than its base

Authored explanation

Cantor’s diagonal argument proves that $|X|<|\mathcal P(X)|$ for every set $X$ .

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
Open Logic Project — Set Theory — Open Logic Project: Set Theory · open textbook source · source ID openlogic-set-theory

Power set Cardinality of the continuum $\mathfrak c$

Permalink to relation

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

Power set Cumulative hierarchy

Permalink to relation

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

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

Authored explanation

The successor clause $V_{\alpha+1}=\mathcal P(V_\alpha)$ adds every subset of the preceding stage; limit stages instead take unions of all earlier stages.

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

Open Logic Project — Set Theory — Open Logic Project: Set Theory · open textbook source · source ID openlogic-set-theory
Stanford Encyclopedia of Philosophy — Set Theory — Set Theory · scholarly reference · source ID sep-set-theory

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Power set

Summary

Record metadata

Carrier(s)

Data

Canonically induces

Notes

Concept sources

Incoming relations (arrows to this concept)

Set Power set

Authored explanation

How to interpret this relation type

Relation sources

Zermelo–Fraenkel set theory (ZF)Power set

Authored explanation

How to interpret this relation type

Relation sources

Outgoing relations (arrows from this concept)

Power set Beth hierarchy

Authored explanation

How to interpret this relation type

Relation sources

Power set Cantor’s theorem

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

Power set Cumulative hierarchy

Authored explanation

How to interpret this relation type

Relation sources