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Inaccessible cardinal

A strongly inaccessible cardinal is an uncountable regular strong-limit cardinal; in modern usage, the unqualified term inaccessible cardinal normally denotes this strong form.

Version
v1 · 2026-09-28 · History
Domain-specific #
10013
Domain group
Formal Sciences
Origin domain
Mathematics
Subdomains
Set Theory, Large Cardinals → Mathematics

Core Idea

Inaccessible cardinal is treated here as the recurring mathematicslogicstatistics identity summarized by this source-grounded definition: A strongly inaccessible cardinal is an uncountable regular strong-limit cardinal; in modern usage, the unqualified term inaccessible cardinal normally denotes this strong form. In set theory, a cardinal number is a strongly inaccessible cardinal if it is uncountable, regular, and a strong limit cardinal. A cardinal is a weakly inaccessible cardinal if it is uncountable, regular, and a weak limit cardinal. Since about 1950, "inaccessible cardinal" has typically meant "strongly inaccessible cardinal" whereas before it had meant "weakly inaccessible cardinal".

Scope of Application

  • Existence of a proper class of inaccessibles. The axioms of ZFC along with the universe axiom (or equivalently the inaccessible cardinal axiom) are denoted ZFCU (not to be confused with ZFC with urelements).

  • Existence of a proper class of inaccessibles. The \alpha -inaccessible cardinals can also be described as fixed points of functions which count the lower inaccessibles.

  • Existence of a proper class of inaccessibles. This process of taking fixed points of functions generating successively larger cardinals is commonly encountered in the study of large cardinal numbers.

  • Existence of a proper class of inaccessibles. (It can never be \kappa + 1 -inaccessible.) It is occasionally used to mean Mahlo cardinal.

  • Models and consistency. Zermelo–Fraenkel set theory with Choice (ZFC) implies that the \kappa th level of the Von Neumann universe V\kappa is a model of ZFC whenever \kappa is strongly inaccessible.

Clarity

A clear use of Inaccessible cardinal names the carrier, the operative relation, and the conditions under which the source treats the identity as present. The minimal definition is In set theory, a cardinal number is a strongly inaccessible cardinal if it is uncountable, regular, and a strong limit cardinal.

Manages Complexity

Inaccessible cardinal compresses multiple mathematicslogicstatistics details into a stable diagnostic relation. The source shows both the central mechanism—for example, denote by \psi0(\lambda) the \lambda th inaccessible cardinal, then the fixed points of \psi0 are the 1-inaccessible cardinals.—and the practical consequence—strongly inaccessible cardinals were introduced by and ; in the latter they were referred to along with \aleph0 as Grenzzahlen (English "limit numbers").

Abstract Reasoning

  1. Type the carrier. Identify the mathematicslogicstatistics entities to which the claim applies.
  2. State the relation. Use the source-grounded identity: In set theory, a cardinal number is a strongly inaccessible cardinal if it is uncountable, regular, and a strong limit cardinal.
  3. Check operation and conditions. This process of taking fixed points of functions generating successively larger cardinals is commonly encountered in the study of large cardinal numbers.
  4. Demand recognition evidence.

Knowledge Transfer

Within the home domain. Knowledge about Inaccessible cardinal transfers literally when a new case preserves the same carrier type, relation, and recognition test. The axioms of ZFC along with the universe axiom (or equivalently the inaccessible cardinal axiom) are denoted ZFCU (not to be confused with ZFC with urelements). The \alpha -inaccessible cardinals can also be described as fixed points of functions which count the lower inaccessibles. Beyond the home domain. No canonical parent is asserted for Inaccessible cardinal.

Relationships to Other Abstractions

Local relationship map for Inaccessible cardinalParents appear above the current abstraction, mutual partners to the right, and children below. Node labels state whether each abstraction is prime or domain-specific; colors identify relation types.Inaccessible cardinalDOMAINPrime abstraction: Cardinality — is a kind ofCardinalityPRIMEDomain-specific abstraction: Huge Cardinal — is a kind ofHuge CardinalDOMAIN

Current abstraction Inaccessible cardinal Domain-specific

Parents (1) — more general patterns this builds on

  • Inaccessible cardinal is a kind of Cardinality Prime

    An inaccessible cardinal is a cardinal number distinguished by uncountability, regularity, and strong-limit conditions; Cardinal Number is a declared alias of the live Cardinality Prime.

Children (1) — more specific cases that build on this

  • Huge Cardinal Domain-specific is a kind of Inaccessible cardinal

    A huge cardinal is strongly inaccessible under ZFC at the same critical-point cardinal.

Hierarchy paths (5) — routes to 3 parentless roots

Neighborhood in Abstraction Space

Inaccessible cardinal sits in a moderately populated region (43rd percentile for distinctiveness): it has near-neighbors but no dense thicket of look-alikes.

Family — Named Analytic Theorems & Operators (39 abstractions)

Nearest neighbors

Computed from structural-signature embeddings · 2026-10-08