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Canonical static concept record

Cantor’s theorem

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Summary

For every set $X$ , no function $X\to\mathcal P(X)$ is surjective; hence $|X|<|\mathcal P(X)|$ .

Record metadata

Canonical name: Cantor’s theorem
Concept ID: cantors_theorem
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: Theorem

Carrier(s)

set $X$

Data

diagonal subset $D=\{x\in X:x\notin f(x)\}$

Axioms / constraints

ordinary classical or intuitionistic set theory suffices

Canonically induces

an unending hierarchy of larger cardinalities

Notes

The theorem rules out a greatest cardinal and does not depend on the continuum hypothesis.

Concept sources

Wikipedia — Cantor’s theorem — Cantor’s theorem · encyclopedic reference · source ID wp-cantor-theorem
nLab — Cantor's theorem — Cantor's theorem · mathematical reference · source ID nlab-cantors-theorem
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.

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

Outgoing relations (arrows from this concept)

Each relation below starts at this concept.

Cantor’s theorem Beth hierarchy

Permalink to relation

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

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Cantor’s theorem

Summary

Record metadata

Carrier(s)

Data

Axioms / constraints

Canonically induces

Notes

Concept sources

Incoming relations (arrows to this concept)

Power set Cantor’s theorem

Authored explanation

How to interpret this relation type

Relation sources

Outgoing relations (arrows from this concept)

Cantor’s theorem Beth hierarchy

Authored explanation

How to interpret this relation type

Relation sources