Inaccessible cardinal

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 has 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 ${\displaystyle \aleph _{0}}$ 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 ${\displaystyle \kappa }$ describe a cardinality ${\displaystyle \kappa }$ which can not be obtained as the cardinality of a result of typical set-theoretic operations involving only sets of cardinality less than ${\displaystyle \kappa }$. Hence the word "inaccessible". By mandating that inaccessible cardinals are uncountable, they turn out to be very large.

In particular, inaccessible cardinals need not exist at all. That is, it is believed that there are models of Zermelo-Fraenkel set theory, even with the axiom of choice (ZFC), for which no inaccessible cardinals exist. On the other hand, it also believed that there are models of ZFC for which even strongly inaccessible cardinals do exist. That ZFC can accommodate these large sets, but does not necessitate them, provides an introduction to the large cardinal axioms. See also Models and consistency.

The existence of a strongly inaccessible cardinal is equivalent to the existence of a Grothendieck universe. If ${\displaystyle \kappa }$ is a strongly inaccessible cardinal then the von Neumann stage ${\displaystyle V_{\kappa }}$ is a Grothendieck universe. Conversely, if ${\displaystyle U}$ is a Grothendieck universe then there is a strongly inaccessible cardinal ${\displaystyle \kappa }$ such that ${\displaystyle V_{\kappa }=U}$. As expected from their correspondence with strongly inaccessible cardinals, Grothendieck universes are very well-closed under set-theoretic operations.

An ordinal is a weakly inaccessible cardinal if and only if it is a regular ordinal and it is a limit of regular ordinals. (Zero, one, and ω are regular ordinals, but not limits of regular ordinals.)

From some perspectives, the requirement that a weakly or strongly inaccessible cardinal be uncountable is unnatural or unnecessary. Even though ⁠${\displaystyle \aleph _{0}}$⁠ is countable, it is regular and is a strong limit cardinal. ⁠${\displaystyle \aleph _{0}}$⁠ is also the smallest weak limit regular cardinal. Assuming the axiom of choice, every other infinite cardinal number is either regular or a weak limit cardinal. However, only a rather large cardinal number can be both. Since a cardinal ⁠${\displaystyle \kappa }$⁠ larger than ⁠${\displaystyle \aleph _{0}}$⁠ is necessarily uncountable, if ⁠${\displaystyle \kappa }$⁠ is also regular and a weak limit cardinal then ⁠${\displaystyle \kappa }$⁠ must be a weakly inaccessible cardinal.