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.
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. The word weakly distinguishes it from strongly inaccessible cardinals, which also satisfy 2^λ < κ for every λ<κ.
Scope of Application¶
Use the term under a stated set theory and keep regularity, cardinal-limit, and strong-limit conditions distinct. 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. The closest near miss sets the boundary: A strongly inaccessible cardinal is closest: it satisfies all weak conditions plus the strong-limit requirement. A positive case must satisfy this test: A cardinal is weakly inaccessible exactly when it is uncountable, regular, and a limit cardinal.
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. The central weak definition–modern terminology tradeoff is this: Historical 'inaccessible' can mean the weak notion while modern unqualified use usually means strong. A second formal definition–unprovable existence tension matters because Properties are elementary to state while existence exceeds ordinary ZFC proof strength.
Abstract Reasoning¶
Use three linked moves: verify κ is a cardinal and uncountable; compute or establish cofinality κ; show κ is not a successor cardinal. As a collapse test, the case exits when kappa is countable, singular, or a successor cardinal. A fourth check is to check separately whether κ is a strong limit. A final check is to 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. No canonical parent prime is currently asserted; broader structural comparisons remain related-prime analogies until separately adjudicated in the DAG. Cofinality equals the cardinal. The cardinal is not a successor.
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
- Stationary set — 0.87
- Accessible category — 0.86
- Inaccessible cardinal — 0.84
- FinSet — 0.83
- Second-Order Predicate — 0.83
Computed from structural-signature embeddings · 2026-10-08