Inaccessible cardinal

Common card · No. 322181

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". Weakly inaccessible cardinals were introduced by Hausdorff (1908). Strongly inaccessible cardinals were introduced by Sierpiński & Tarski (1930) and Zermelo (1930); in the latter they were referred to along with ℵ 0 {\displaystyle \aleph _{0}} as Grenzzahlen (English "limit numbers"). Every strongly inaccessible cardinal is a weakly inaccessible cardinal. The generalized continuum hypothesis implies that all weakly inaccessible cardinals are strongly inaccessible as well. The two notions of an inaccessible cardinal...

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