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.

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 . (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.)

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.

Infinite well-ordered cardinals which are not regular are called **singular cardinals**. Finite cardinal numbers are typically not called regular or singular.

Read more about Regular Cardinal: Examples, Properties

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### Famous quotes containing the words cardinal and/or regular:

“Time and I against any two.”

—Spanish proverb.

Quoted by *Cardinal* Mazarin during the minority of Louis XIV.

“He hung out of the window a long while looking up and down the street. The world’s second metropolis. In the brick houses and the dingy lamplight and the voices of a group of boys kidding and quarreling on the steps of a house opposite, in the *regular* firm tread of a policeman, he felt a marching like soldiers, like a sidewheeler going up the Hudson under the Palisades, like an election parade, through long streets towards something tall white full of colonnades and stately. Metropolis.”

—John Dos Passos (1896–1970)