Regular cardinal

In set theory, a regular cardinal is a cardinal number that is equal to its own cofinality. So, crudely speaking, a regular cardinal is one that cannot be broken into a smaller collection of smaller parts.

If the axiom of choice holds (so that any cardinal number can be well-ordered), an infinite cardinal $\kappa$ is regular if and only if it cannot be expressed as the cardinal sum of a set of cardinality less than $\kappa$ , the elements of which are cardinals less than $\kappa$ . (The situation is slightly more complicated in contexts where the axiom of choice might fail; in that case not all cardinals are necessarily the cardinalities of well-ordered sets. In that case, the above definition is restricted to well-orderable cardinals only.)

单词	Regular cardinal
释义	Regular cardinal 英语百科 Regular cardinal In set theory, a regular cardinal is a cardinal number that is equal to its own cofinality. So, crudely speaking, a regular cardinal is one that cannot be broken into a smaller collection of smaller parts. If the axiom of choice holds (so that any cardinal number can be well-ordered), an infinite cardinal $\kappa$ is regular if and only if it cannot be expressed as the cardinal sum of a set of cardinality less than $\kappa$ , the elements of which are cardinals less than $\kappa$ . (The situation is slightly more complicated in contexts where the axiom of choice might fail; in that case not all cardinals are necessarily the cardinalities of well-ordered sets. In that case, the above definition is restricted to well-orderable cardinals only.)
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