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Canonical static concept record

Measurable cardinal

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Summary

An uncountable cardinal $\kappa$ carrying a nonprincipal $\kappa$ -complete ultrafilter.

Record metadata

Canonical name: Measurable cardinal
Concept ID: measurable_cardinal
Content version: 2.3.0
Primary field: Mathematics
Fields: Mathematics
Primary domain: Set theory
Domains: Set theory, Logic, Measure
Alternate terminology: No alternate terminology is currently authored for this record.
Concept type: Large-cardinal notion

Carrier(s)

cardinal $\kappa$

Data

nonprincipal $\kappa$ -complete ultrafilter on $\kappa$

Canonically induces

elementary embedding of the universe into an inner model

Notes

Every measurable cardinal is strongly inaccessible in ZFC.

Concept sources

Wikipedia — Measurable cardinal — Measurable cardinal · encyclopedic reference · source ID wp-measurable-cardinal
nLab — Measurable cardinal — Measurable cardinal · mathematical reference · source ID nlab-measurable-cardinal
Stanford Encyclopedia of Philosophy — Large Cardinals and Determinacy — Large Cardinals and Determinacy · scholarly reference · source ID sep-large-cardinals

Incoming relations (arrows to this concept)

Each relation below ends at this concept.

Large cardinal Measurable cardinal

Permalink to relation

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

Relation ID: large_cardinal_to_measurable
Relation type: Classification classification
Direction: source → target
Endpoint roles: source: Has subtype; target: Classified as
Authored annotation: measurable cardinals

Authored explanation

Measurable cardinals are a stronger large-cardinal notion defined by a highly complete nonprincipal measure.

How to interpret this relation type

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

Relation sources

Wikipedia — Measurable cardinal — Measurable cardinal · encyclopedic reference · source ID wp-measurable-cardinal
Stanford Encyclopedia of Philosophy — Large Cardinals and Determinacy — Large Cardinals and Determinacy · scholarly reference · source ID sep-large-cardinals

Supercompact cardinal Measurable cardinal

Permalink to relation

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

Relation ID: supercompact_implies_measurable
Relation type: Theorem implication theorem-implication
Direction: source → target
Endpoint roles: source: Implies by theorem; target: Follows by theorem from
Authored annotation: supercompact implies measurable

Authored explanation

Every supercompact cardinal is measurable.

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 — Supercompact cardinal — Supercompact cardinal · encyclopedic reference · source ID wp-supercompact-cardinal
Wikipedia — Measurable cardinal — Measurable cardinal · encyclopedic reference · source ID wp-measurable-cardinal

Outgoing relations (arrows from this concept)

Each relation below starts at this concept.