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Weakly Inaccessible Cardinal

An uncountable regular limit cardinal: a cardinal greater than aleph-zero whose cofinality equals itself and which is not the successor of any smaller cardinal, without requiring the strong-limit property.

Version
v1 · 2026-09-28 · History
Domain-specific #
12851
Domain group
Formal Sciences
Origin domain
Mathematics
Subdomains
Set Theory, Large Cardinals → Mathematics

Core Idea

A weakly inaccessible cardinal κ is an uncountable regular limit cardinal. Regularity says no smaller cofinal sequence reaches κ; limit-cardinal status says κ is not the immediate cardinal successor of a smaller size. These requirements are independent checks: neither regularity nor limit status alone establishes the definition.

The word weakly distinguishes it from strongly inaccessible cardinals, which also satisfy 2^λ < κ for every λ<κ. Every strong inaccessible is weakly inaccessible, but the converse need not hold without additional assumptions; GCH makes the two notions coincide. ZFC alone does not establish existence if it is consistent.

Structural Signature

Sig role-phrases:

  • cardinal kappa. Supplies the size under classification. Constitutive object. If altered: An ordinal not treated as a cardinal is outside the definition.
  • uncountability. Requires kappa exceed aleph-zero. Constitutive lower bound. If altered: Aleph-zero is regular and limit-like but excluded by convention.
  • regularity. Requires cofinality(kappa)=kappa. Constitutive closure property. If altered: A singular limit cardinal is not weakly inaccessible.
  • limit-cardinal status. Requires kappa not be a successor cardinal. Identity-bearing weak-limit condition. If altered: A regular successor cardinal fails despite size.
  • strong-limit distinction. Leaves open whether 2^lambda<kappa for every smaller lambda. Necessary boundary from strong inaccessibility. If altered: Adding it yields the strong notion.

What It Is Not

  • Strongly inaccessible cardinal. Is strong-limit arithmetic required?
  • Singular limit cardinal. Does regularity fail?
  • Limit ordinal. Is cardinal-limit status established?
  • Regular successor cardinal. Is the size merely a successor?

Scope of Application

Use the term under a stated set theory and keep regularity, cardinal-limit, and strong-limit conditions distinct.

  • Large-cardinal theory. Locates a low inaccessible notion.
  • Set-theoretic universes. Studies closure at V-kappa.
  • Cardinal arithmetic. Tests strong-limit dependence.
  • Model theory. States consistency strength carefully.
  • History of terminology. Interprets older 'inaccessible' usage.

Clarity

Limit cardinal is not merely limit ordinal, and regularity is not implied by being a limit. All three clauses must be checked.

Manages Complexity

The definition is short but existence has metamathematical weight. Statements should distinguish theorem in ZFC, conditional implication such as GCH, and consistency assumptions.

Abstract Reasoning

  1. Verify κ is a cardinal and uncountable.
  2. Compute or establish cofinality κ.
  3. Show κ is not a successor cardinal.
  4. Check separately whether κ is a strong limit.
  5. State background axioms and existence assumptions.

Knowledge Transfer

Regular limit structure transfers analogically to other orders, but cardinality, cofinality, and set-theoretic axioms delimit the concept. The nearest stopping boundary is explicit: A strongly inaccessible cardinal is closest: it satisfies all weak conditions plus the strong-limit requirement. The inclusion test remains: A cardinal is weakly inaccessible exactly when it is uncountable, regular, and a limit cardinal. The structure no longer applies when the case exits when kappa is countable, singular, or a successor cardinal.

Examples

Canonical

A hypothetical cardinal κ is greater than aleph-zero, has no cofinal sequence of length below κ, and is a limit of smaller cardinals; it is weakly inaccessible whether or not it is a strong limit.

Mapped back: cardinal kappa → κ; uncountability → κ>aleph0; regularity → cf κ=κ; limit-cardinal status → not successor; strong-limit distinction → not assumed.

Applied / In Practice

A singular limit cardinal is uncountable and not a successor, but a smaller cofinal sequence reaches it; failure of regularity excludes weak inaccessibility.

Mapped back: cardinal kappa → singular limit; uncountability → present; regularity → fails; limit-cardinal status → present; strong-limit distinction → irrelevant.

Structural Tensions

T1: weak definition vs. modern terminology. Historical 'inaccessible' can mean the weak notion while modern unqualified use usually means strong. Diagnostic: Which era and definition govern the text?

T2: formal definition vs. unprovable existence. Properties are elementary to state while existence exceeds ordinary ZFC proof strength. Diagnostic: Is the claim conditional or existential?

Structural–Framed Character

Description turns on cardinal kappa, uncountability, regularity, limit-cardinal status, strong-limit distinction. Skeletal core. An ordered size is both a non-successor limit and internally cofinal only at its own scale. Domain-bound accent. Cardinals, cofinality, successors, strong limits, GCH, and ZFC define weak inaccessibility. Transfer remains bounded because Why not prime. Limit regularity is portable only analogically; this is a set-theoretic large-cardinal property. The negative boundary is concrete: Any large cardinal, uncountable cardinal, regular cardinal, limit ordinal, singular limit cardinal, successor cardinal, or strongly inaccessible cardinal label is not automatically the weak definition, though strong inaccessibles form a subclass. Weak inaccessibility is structural-formal: cardinal, cofinality, and successor relations determine it exactly. Its character: uncountable cardinal height unreachable by successor and smaller cofinal buildup.

Structural Core vs. Domain Accent

Skeletal core. An ordered size is both a non-successor limit and internally cofinal only at its own scale.

Domain-bound accent. Cardinals, cofinality, successors, strong limits, GCH, and ZFC define weak inaccessibility.

Why not prime. Limit regularity is portable only analogically; this is a set-theoretic large-cardinal property.

  • Regularity. Cofinality equals the cardinal.
  • Limit. The cardinal is not a successor.
  • No strict parent is asserted.

Neighborhood in Abstraction Space

Weakly Inaccessible Cardinal sits in a sparse region of the domain-specific corpus (68th percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.

Family — Abstract Algebra & Category Theory (24 abstractions)

Nearest neighbors

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

Not to Be Confused With

  • Strongly inaccessible cardinal. Tell: Is strong-limit arithmetic required?
  • Singular limit cardinal. Tell: Does regularity fail?
  • Limit ordinal. Tell: Is cardinal-limit status established?
  • Regular successor cardinal. Tell: Is the size merely a successor?

References

  • Frozen Wikipedia discovery revision: https://en.wikipedia.org/wiki/Inaccessible_cardinal (revision 1369414153).
  • Preserved source candidate: https://math.bu.edu/people/aki/10.pdf
  • Preserved source candidate: https://mathoverflow.net/questions/437195/does-anyone-still-seriously-doubt-the-consistency-of-zfc
  • Preserved source candidate: https://www.jstor.org/stable/26788522
  • Preserved source candidate: https://gdz.sub.uni-goettingen.de/id/PPN235181684_0065?tify={%22pages%22:[453
  • Preserved source candidate: http://matwbn.icm.edu.pl/ksiazki/fm/fm15/fm15129.pdf
  • Preserved source candidate: http://matwbn.icm.edu.pl/ksiazki/fm/fm16/fm1615.pdf

The frozen Wikipedia revision is discovery provenance. The retained source set was reviewed for identity, formal or operational relation, and scope. The encyclopedia's structural synthesis is bounded to those claims; a thin authority surface is recorded as a nonblocking source-strengthening repair rather than concealed.