cardinal is inaccessible if it cannot be obtained from smaller cardinals by the usual operations of cardinal arithmetic. More precisely, a cardinal κ...

A cardinal κ {\displaystyle \kappa } is called weakly Mahlo if κ {\displaystyle \kappa } is weakly inaccessible and the set of weakly inaccessible cardinals...

worldly cardinal is a cardinal κ such that the rank Vκ is a model of Zermelo–Fraenkel set theory. By Zermelo's theorem on inaccessible cardinals, every...

Grothendieck universe U such that x ∈ U. (C) For each cardinal κ, there is a strongly inaccessible cardinal λ that is strictly larger than κ. To prove this...

not inaccessible. ℵ 0 {\displaystyle \aleph _{0}} would be an inaccessible cardinal of both "strengths" except that the definition of inaccessible requires...

(see Aleph number) worldly cardinals weakly and strongly inaccessible, α-inaccessible, and hyper inaccessible cardinals weakly and strongly Mahlo, α-Mahlo...

inaccessible cardinal, then "cutting the universe off" at the height of the first such cardinal yields a universe in which there is no inaccessible cardinal. Or...

itself, but ZFC + "there exists an inaccessible cardinal" proves ZFC is consistent because if κ is the least such cardinal, then Vκ sitting inside the von...

hyper-inaccessible 1.  "Hyper-inaccessible cardinal" usually means a 1-inaccessible cardinal 2.  "Hyper-inaccessible cardinal" sometimes means a cardinal κ...

smaller large cardinals, such as inaccessible cardinals. (Roughly speaking, the class of all ordinals in ZFC is an inaccessible cardinal apart from the...

weakly inaccessible cardinal is also a fixed point of the aleph function. This can be shown in ZFC as follows. Suppose κ = ℵλ is a weakly inaccessible cardinal...

set-theoretic properties, for example when κ {\displaystyle \kappa } is an inaccessible cardinal, V κ {\displaystyle V_{\kappa }} satisfies second-order ZFC ("satisfies"...

this axiom. Uncountable (weak) limit cardinals that are also regular are known as (weakly) inaccessible cardinals. They cannot be proved to exist within...

Lebesgue measurable. The construction relies on the existence of an inaccessible cardinal. In this way Solovay showed that in the proof of the existence of...

theory for supercompact cardinals is developed. Jech obtained a variant of the tree property which holds for an inaccessible cardinal if and only if it is...

properties that define strongly inaccessible cardinals. A cardinal κ is strongly inaccessible if κ > ω and: If λ is a cardinal such that λ < κ, then 2λ < κ...

measurable. However, Solovay's result depends on the existence of an inaccessible cardinal, whose existence and consistency cannot be proved within standard...

Ramsey cardinal Erdős cardinal Extendible cardinal Huge cardinal Hyper-Woodin cardinal Inaccessible cardinal Ineffable cardinal Mahlo cardinal Measurable...

every set S of cardinality κ of subsets of κ, there is a non-trivial κ-complete filter that decides S. κ is κ-unfoldable. κ is inaccessible and the infinitary...

The consistency of ZFC does follow from the existence of a weakly inaccessible cardinal, which is unprovable in ZFC if ZFC is consistent. Nevertheless,...

needing the axiom of replacement to go outside Vω+ω. If κ is an inaccessible cardinal, then Vκ is a model of Zermelo–Fraenkel set theory (ZFC) itself...

Non-existence of a two-valued σ-measure for the first uncountable inaccessible cardinal", Acta Mathematica Academiae Scientiarum Hungaricae, 13 (1–2): 223–226...

and any strongly inaccessible ethereal cardinal is subtle.p. 391 Some equivalent properties to subtlety are known. Subtle cardinals are equivalent to...

strongly inaccessible cardinal is Lévy collapsed to ω2 then, in the resulting model, there are no Kurepa trees. The existence of an inaccessible cardinal is...

which holds for an inaccessible cardinal iff it is supercompact. Indestructibility Strongly compact cardinal List of large cardinal properties Drake, F...

limit cardinal, which completes the proof of its inaccessibility. Although it follows from ZFC that every measurable cardinal is inaccessible (and is...

reflecting cardinals. The consistency strength of an inaccessible reflecting cardinal is strictly greater than a greatly Mahlo cardinal, where a cardinal κ is...

axioms of ZF do not immediately apply to classes. However, if an inaccessible cardinal κ {\displaystyle \kappa } is assumed, then the sets of smaller rank...

proved in ZFC itself, but requires a mild large cardinal assumption (the existence of an inaccessible cardinal). The much stronger axiom of determinacy, or...

inaccessible cardinals Existence of Mahlo cardinals Existence of measurable cardinals (first conjectured by Ulam) Existence of supercompact cardinals...