cardinal number is a strongly inaccessible cardinal if it is uncountable, regular, and a strong limit cardinal. A cardinal is a weakly inaccessible cardinal...

strongly inaccessible, and for any unbounded set S ⊆ κ {\displaystyle S\subseteq \kappa } of cardinals, there is a strongly inaccessible cardinal λ < κ {\displaystyle...

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...

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

Zermelo's categoricity theorem, every inaccessible cardinal is worldly. By Shepherdson's theorem, inaccessibility is equivalent to the stronger statement...

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...

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

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...

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

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...

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...

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...

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...

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,...

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

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

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...

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

denoted ω ω ⋱ {\textstyle \omega _{\omega _{\ddots }}} . Any weakly inaccessible cardinal is also a fixed point of the aleph function. This can be shown in...

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

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

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...

Cardinal and Ordinal Numbers Cardinal function Inaccessible cardinal Infinitary combinatorics Large cardinal List of large cardinal properties Numerosity (mathematics)...

that are not Lebesgue measurable? The answer is yes, provided that inaccessible cardinals are consistent with the most common axiomatization of set theory...

cardinal is a cardinal number with an extra property. Many such properties are studied, including inaccessible cardinals, measurable cardinals, and many more...

theories such as Morse–Kelley set theory or set theory with a strongly inaccessible cardinal allowing the use of a Grothendieck universe is used, but in fact...