Regular Cardinal
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Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. 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 which cannot be broken into a smaller collection of smaller parts. (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.) ...
Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. 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 which cannot be broken into a smaller collection of smaller parts. (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.) If the axiom of choice holds (so that any cardinal number can be well-ordered), an infinite cardinal is regular if and only if it cannot be expressed as the cardinal sum of a set of cardinality less than , the elements of which are cardinals less than . An infinite ordinal is regular if and only if it is a limit ordinal which is not the limit of a set of smaller ordinals which set has order type less than . A regular ordinal is always an initial ordinal, though some initial ordinals are not regular.
